Algebra Examples

$f (x) = \sqrt[7]{8 x} + 5$

Step 1

Write $f (x) = \sqrt[7]{8 x} + 5$ as an equation.

$y = \sqrt[7]{8 x} + 5$

Step 2

Interchange the variables.

$x = \sqrt[7]{8 y} + 5$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[7]{8 y} + 5 = x$ .

$\sqrt[7]{8 y} + 5 = x$

Step 3.2

Subtract $5$ from both sides of the equation.

$\sqrt[7]{8 y} = x - 5$

Step 3.3

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $7$ .

${\sqrt[7]{8 y}}^{7} = {(x - 5)}^{7}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{8 y}$ as ${(8 y)}^{\frac{1}{7}}$ .

${({(8 y)}^{\frac{1}{7}})}^{7} = {(x - 5)}^{7}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${({(8 y)}^{\frac{1}{7}})}^{7}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${({(8 y)}^{\frac{1}{7}})}^{7}$ .

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Step 3.4.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(8 y)}^{\frac{1}{7} \cdot 7} = {(x - 5)}^{7}$

Step 3.4.2.1.1.2

Cancel the common factor of $7$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

${(8 y)}^{\frac{1}{7} \cdot 7} = {(x - 5)}^{7}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

${(8 y)}^{1} = {(x - 5)}^{7}$

Step 3.4.2.1.2

Simplify.

$8 y = {(x - 5)}^{7}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x - 5)}^{7}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$8 y = x^{7} + 7 x^{6} \cdot - 5 + 21 x^{5} {(- 5)}^{2} + 35 x^{4} {(- 5)}^{3} + 35 x^{3} {(- 5)}^{4} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $- 5$ by $7$ .

$8 y = x^{7} - 35 x^{6} + 21 x^{5} {(- 5)}^{2} + 35 x^{4} {(- 5)}^{3} + 35 x^{3} {(- 5)}^{4} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.2

Raise $- 5$ to the power of $2$ .

$8 y = x^{7} - 35 x^{6} + 21 x^{5} \cdot 25 + 35 x^{4} {(- 5)}^{3} + 35 x^{3} {(- 5)}^{4} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.3

Multiply $25$ by $21$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} + 35 x^{4} {(- 5)}^{3} + 35 x^{3} {(- 5)}^{4} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.4

Raise $- 5$ to the power of $3$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} + 35 x^{4} \cdot - 125 + 35 x^{3} {(- 5)}^{4} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.5

Multiply $- 125$ by $35$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 35 x^{3} {(- 5)}^{4} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.6

Raise $- 5$ to the power of $4$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 35 x^{3} \cdot 625 + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.7

Multiply $625$ by $35$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} + 21 x^{2} {(- 5)}^{5} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.8

Raise $- 5$ to the power of $5$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} + 21 x^{2} \cdot - 3125 + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.9

Multiply $- 3125$ by $21$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 7 x {(- 5)}^{6} + {(- 5)}^{7}$

Step 3.4.3.1.2.10

Raise $- 5$ to the power of $6$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 7 x \cdot 15625 + {(- 5)}^{7}$

Step 3.4.3.1.2.11

Multiply $15625$ by $7$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 109375 x + {(- 5)}^{7}$

Step 3.4.3.1.2.12

Raise $- 5$ to the power of $7$ .

$8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 109375 x - 78125$

Step 3.5

Divide each term in $8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 109375 x - 78125$ by $8$ and simplify.

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Step 3.5.1

Divide each term in $8 y = x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 109375 x - 78125$ by $8$ .

$\frac{8 y}{8} = \frac{x^{7}}{8} + \frac{- 35 x^{6}}{8} + \frac{525 x^{5}}{8} + \frac{- 4375 x^{4}}{8} + \frac{21875 x^{3}}{8} + \frac{- 65625 x^{2}}{8} + \frac{109375 x}{8} + \frac{- 78125}{8}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $8$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{8 y}{8} = \frac{x^{7}}{8} + \frac{- 35 x^{6}}{8} + \frac{525 x^{5}}{8} + \frac{- 4375 x^{4}}{8} + \frac{21875 x^{3}}{8} + \frac{- 65625 x^{2}}{8} + \frac{109375 x}{8} + \frac{- 78125}{8}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{7}}{8} + \frac{- 35 x^{6}}{8} + \frac{525 x^{5}}{8} + \frac{- 4375 x^{4}}{8} + \frac{21875 x^{3}}{8} + \frac{- 65625 x^{2}}{8} + \frac{109375 x}{8} + \frac{- 78125}{8}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify each term.

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Step 3.5.3.1.1

Move the negative in front of the fraction.

$y = \frac{x^{7}}{8} - \frac{(35) x^{6}}{8} + \frac{525 x^{5}}{8} + \frac{- 4375 x^{4}}{8} + \frac{21875 x^{3}}{8} + \frac{- 65625 x^{2}}{8} + \frac{109375 x}{8} + \frac{- 78125}{8}$

Step 3.5.3.1.2

Move the negative in front of the fraction.

$y = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{(4375) x^{4}}{8} + \frac{21875 x^{3}}{8} + \frac{- 65625 x^{2}}{8} + \frac{109375 x}{8} + \frac{- 78125}{8}$

Step 3.5.3.1.3

Move the negative in front of the fraction.

$y = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{(65625) x^{2}}{8} + \frac{109375 x}{8} + \frac{- 78125}{8}$

Step 3.5.3.1.4

Move the negative in front of the fraction.

$y = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}$ is the inverse of $f (x) = \sqrt[7]{8 x} + 5$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (\sqrt[7]{8 x} + 5)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{{(\sqrt[7]{8 x} + 5)}^{7}}{8} - \frac{35 {(\sqrt[7]{8 x} + 5)}^{6}}{8} + \frac{525 {(\sqrt[7]{8 x} + 5)}^{5}}{8} - \frac{4375 {(\sqrt[7]{8 x} + 5)}^{4}}{8} + \frac{21875 {(\sqrt[7]{8 x} + 5)}^{3}}{8} - \frac{65625 {(\sqrt[7]{8 x} + 5)}^{2}}{8} + \frac{109375 (\sqrt[7]{8 x} + 5)}{8} - \frac{78125}{8}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{{(\sqrt[7]{8 x} + 5)}^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{{\sqrt[7]{8 x}}^{7} + 7 {\sqrt[7]{8 x}}^{6} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2

Simplify each term.

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Step 5.2.4.2.1

Rewrite ${\sqrt[7]{8 x}}^{7}$ as $8 x$ .

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Step 5.2.4.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{8 x}$ as ${(8 x)}^{\frac{1}{7}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{{({(8 x)}^{\frac{1}{7}})}^{7} + 7 {\sqrt[7]{8 x}}^{6} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{{(8 x)}^{\frac{1}{7} \cdot 7} + 7 {\sqrt[7]{8 x}}^{6} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.1.3

Combine $\frac{1}{7}$ and $7$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{{(8 x)}^{\frac{7}{7}} + 7 {\sqrt[7]{8 x}}^{6} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.1.4

Cancel the common factor of $7$ .

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Step 5.2.4.2.1.4.1

Cancel the common factor.

Step 5.2.4.2.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{(8 x) + 7 {\sqrt[7]{8 x}}^{6} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.1.5

Simplify.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 {\sqrt[7]{8 x}}^{6} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.2

Rewrite ${\sqrt[7]{8 x}}^{6}$ as $\sqrt[7]{{(8 x)}^{6}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 \sqrt[7]{{(8 x)}^{6}} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.3

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 \sqrt[7]{8^{6} x^{6}} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.4

Raise $8$ to the power of $6$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 \sqrt[7]{262144 x^{6}} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.5

Rewrite $262144 x^{6}$ as $4^{7} \cdot (16 x^{6})$ .

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Step 5.2.4.2.5.1

Factor $16384$ out of $262144$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 \sqrt[7]{16384 (16) x^{6}} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.5.2

Rewrite $16384$ as $4^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 \sqrt[7]{4^{7} \cdot (16 x^{6})} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.5.3

Add parentheses.

Step 5.2.4.2.6

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 7 (4 \sqrt[7]{16 x^{6}}) \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.7

Multiply $4$ by $7$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 28 \sqrt[7]{16 x^{6}} \cdot 5 + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.8

Multiply $5$ by $28$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 {\sqrt[7]{8 x}}^{5} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.9

Rewrite ${\sqrt[7]{8 x}}^{5}$ as $\sqrt[7]{{(8 x)}^{5}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 \sqrt[7]{{(8 x)}^{5}} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.10

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 \sqrt[7]{8^{5} x^{5}} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.11

Raise $8$ to the power of $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 \sqrt[7]{32768 x^{5}} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.12

Rewrite $32768 x^{5}$ as $4^{7} \cdot (2 x^{5})$ .

Tap for more steps...

Step 5.2.4.2.12.1

Factor $16384$ out of $32768$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 \sqrt[7]{16384 (2) x^{5}} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.12.2

Rewrite $16384$ as $4^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 \sqrt[7]{4^{7} \cdot (2 x^{5})} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.12.3

Add parentheses.

Step 5.2.4.2.13

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 21 (4 \sqrt[7]{2 x^{5}}) \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.14

Multiply $4$ by $21$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 84 \sqrt[7]{2 x^{5}} \cdot 5^{2} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.15

Raise $5$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 84 \sqrt[7]{2 x^{5}} \cdot 25 + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.16

Multiply $25$ by $84$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 {\sqrt[7]{8 x}}^{4} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.17

Rewrite ${\sqrt[7]{8 x}}^{4}$ as $\sqrt[7]{{(8 x)}^{4}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 \sqrt[7]{{(8 x)}^{4}} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.18

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 \sqrt[7]{8^{4} x^{4}} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.19

Raise $8$ to the power of $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 \sqrt[7]{4096 x^{4}} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.20

Rewrite $4096 x^{4}$ as $2^{7} \cdot (32 x^{4})$ .

Tap for more steps...

Step 5.2.4.2.20.1

Factor $128$ out of $4096$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 \sqrt[7]{128 (32) x^{4}} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.20.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 \sqrt[7]{2^{7} \cdot (32 x^{4})} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.20.3

Add parentheses.

Step 5.2.4.2.21

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 35 (2 \sqrt[7]{32 x^{4}}) \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.22

Multiply $2$ by $35$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 70 \sqrt[7]{32 x^{4}} \cdot 5^{3} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.23

Raise $5$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 70 \sqrt[7]{32 x^{4}} \cdot 125 + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.24

Multiply $125$ by $70$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 {\sqrt[7]{8 x}}^{3} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.25

Rewrite ${\sqrt[7]{8 x}}^{3}$ as $\sqrt[7]{{(8 x)}^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 \sqrt[7]{{(8 x)}^{3}} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.26

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 \sqrt[7]{8^{3} x^{3}} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.27

Raise $8$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 \sqrt[7]{512 x^{3}} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.28

Rewrite $512 x^{3}$ as $2^{7} \cdot (4 x^{3})$ .

Tap for more steps...

Step 5.2.4.2.28.1

Factor $128$ out of $512$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 \sqrt[7]{128 (4) x^{3}} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.28.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 \sqrt[7]{2^{7} \cdot (4 x^{3})} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.28.3

Add parentheses.

Step 5.2.4.2.29

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 35 (2 \sqrt[7]{4 x^{3}}) \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.30

Multiply $2$ by $35$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 70 \sqrt[7]{4 x^{3}} \cdot 5^{4} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.31

Raise $5$ to the power of $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 70 \sqrt[7]{4 x^{3}} \cdot 625 + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.32

Multiply $625$ by $70$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 21 {\sqrt[7]{8 x}}^{2} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.33

Rewrite ${\sqrt[7]{8 x}}^{2}$ as $\sqrt[7]{{(8 x)}^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 21 \sqrt[7]{{(8 x)}^{2}} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.34

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 21 \sqrt[7]{8^{2} x^{2}} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.35

Raise $8$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 21 \sqrt[7]{64 x^{2}} \cdot 5^{5} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.36

Raise $5$ to the power of $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 21 \sqrt[7]{64 x^{2}} \cdot 3125 + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.37

Multiply $3125$ by $21$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 7 \sqrt[7]{8 x} \cdot 5^{6} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.38

Raise $5$ to the power of $6$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 7 \sqrt[7]{8 x} \cdot 15625 + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.39

Multiply $15625$ by $7$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 5^{7} - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.2.40

Raise $5$ to the power of $7$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 {(\sqrt[7]{8 x} + 5)}^{6} + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.3

Use the Binomial Theorem.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 ({\sqrt[7]{8 x}}^{6} + 6 {\sqrt[7]{8 x}}^{5} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4

Simplify each term.

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Step 5.2.4.4.1

Rewrite ${\sqrt[7]{8 x}}^{6}$ as $\sqrt[7]{{(8 x)}^{6}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (\sqrt[7]{{(8 x)}^{6}} + 6 {\sqrt[7]{8 x}}^{5} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.2

Apply the product rule to $8 x$ .

Step 5.2.4.4.3

Raise $8$ to the power of $6$ .

Step 5.2.4.4.4

Rewrite $262144 x^{6}$ as $4^{7} \cdot (16 x^{6})$ .

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Step 5.2.4.4.4.1

Factor $16384$ out of $262144$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (\sqrt[7]{16384 (16) x^{6}} + 6 {\sqrt[7]{8 x}}^{5} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.4.2

Rewrite $16384$ as $4^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (\sqrt[7]{4^{7} \cdot (16 x^{6})} + 6 {\sqrt[7]{8 x}}^{5} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.4.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (\sqrt[7]{4^{7} \cdot (16 x^{6})} + 6 {\sqrt[7]{8 x}}^{5} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.5

Pull terms out from under the radical.

Step 5.2.4.4.6

Rewrite ${\sqrt[7]{8 x}}^{5}$ as $\sqrt[7]{{(8 x)}^{5}}$ .

Step 5.2.4.4.7

Apply the product rule to $8 x$ .

Step 5.2.4.4.8

Raise $8$ to the power of $5$ .

Step 5.2.4.4.9

Rewrite $32768 x^{5}$ as $4^{7} \cdot (2 x^{5})$ .

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Step 5.2.4.4.9.1

Factor $16384$ out of $32768$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (4 \sqrt[7]{16 x^{6}} + 6 \sqrt[7]{16384 (2) x^{5}} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.9.2

Rewrite $16384$ as $4^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (4 \sqrt[7]{16 x^{6}} + 6 \sqrt[7]{4^{7} \cdot (2 x^{5})} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.9.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 35 (4 \sqrt[7]{16 x^{6}} + 6 \sqrt[7]{4^{7} \cdot (2 x^{5})} \cdot 5 + 15 {\sqrt[7]{8 x}}^{4} \cdot 5^{2} + 20 {\sqrt[7]{8 x}}^{3} \cdot 5^{3} + 15 {\sqrt[7]{8 x}}^{2} \cdot 5^{4} + 6 \sqrt[7]{8 x} \cdot 5^{5} + 5^{6}) + 525 {(\sqrt[7]{8 x} + 5)}^{5} - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.4.10

Pull terms out from under the radical.

Step 5.2.4.4.11

Multiply $4$ by $6$ .

Step 5.2.4.4.12

Multiply $5$ by $24$ .

Step 5.2.4.4.13

Rewrite ${\sqrt[7]{8 x}}^{4}$ as $\sqrt[7]{{(8 x)}^{4}}$ .

Step 5.2.4.4.14

Apply the product rule to $8 x$ .

Step 5.2.4.4.15

Raise $8$ to the power of $4$ .

Step 5.2.4.4.16

Rewrite $4096 x^{4}$ as $2^{7} \cdot (32 x^{4})$ .

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Step 5.2.4.4.16.1

Factor $128$ out of $4096$ .

Step 5.2.4.4.16.2

Rewrite $128$ as $2^{7}$ .

Step 5.2.4.4.16.3

Add parentheses.

Step 5.2.4.4.17

Pull terms out from under the radical.

Step 5.2.4.4.18

Multiply $2$ by $15$ .

Step 5.2.4.4.19

Raise $5$ to the power of $2$ .

Step 5.2.4.4.20

Multiply $25$ by $30$ .

Step 5.2.4.4.21

Rewrite ${\sqrt[7]{8 x}}^{3}$ as $\sqrt[7]{{(8 x)}^{3}}$ .

Step 5.2.4.4.22

Apply the product rule to $8 x$ .

Step 5.2.4.4.23

Raise $8$ to the power of $3$ .

Step 5.2.4.4.24

Rewrite $512 x^{3}$ as $2^{7} \cdot (4 x^{3})$ .

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Step 5.2.4.4.24.1

Factor $128$ out of $512$ .

Step 5.2.4.4.24.2

Rewrite $128$ as $2^{7}$ .

Step 5.2.4.4.24.3

Add parentheses.

Step 5.2.4.4.25

Pull terms out from under the radical.

Step 5.2.4.4.26

Multiply $2$ by $20$ .

Step 5.2.4.4.27

Raise $5$ to the power of $3$ .

Step 5.2.4.4.28

Multiply $125$ by $40$ .

Step 5.2.4.4.29

Rewrite ${\sqrt[7]{8 x}}^{2}$ as $\sqrt[7]{{(8 x)}^{2}}$ .

Step 5.2.4.4.30

Apply the product rule to $8 x$ .

Step 5.2.4.4.31

Raise $8$ to the power of $2$ .

Step 5.2.4.4.32

Raise $5$ to the power of $4$ .

Step 5.2.4.4.33

Multiply $625$ by $15$ .

Step 5.2.4.4.34

Raise $5$ to the power of $5$ .

Step 5.2.4.4.35

Multiply $3125$ by $6$ .

Step 5.2.4.4.36

Raise $5$ to the power of $6$ .

Step 5.2.4.5

Apply the distributive property.

Step 5.2.4.6

Simplify.

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Step 5.2.4.6.1

Multiply $4$ by $- 35$ .

Step 5.2.4.6.2

Multiply $120$ by $- 35$ .

Step 5.2.4.6.3

Multiply $750$ by $- 35$ .

Step 5.2.4.6.4

Multiply $5000$ by $- 35$ .

Step 5.2.4.6.5

Multiply $9375$ by $- 35$ .

Step 5.2.4.6.6

Multiply $18750$ by $- 35$ .

Step 5.2.4.6.7

Multiply $- 35$ by $15625$ .

Step 5.2.4.7

Use the Binomial Theorem.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 ({\sqrt[7]{8 x}}^{5} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8

Simplify each term.

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Step 5.2.4.8.1

Rewrite ${\sqrt[7]{8 x}}^{5}$ as $\sqrt[7]{{(8 x)}^{5}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (\sqrt[7]{{(8 x)}^{5}} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.2

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (\sqrt[7]{8^{5} x^{5}} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.3

Raise $8$ to the power of $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (\sqrt[7]{32768 x^{5}} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.4

Rewrite $32768 x^{5}$ as $4^{7} \cdot (2 x^{5})$ .

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Step 5.2.4.8.4.1

Factor $16384$ out of $32768$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (\sqrt[7]{16384 (2) x^{5}} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.4.2

Rewrite $16384$ as $4^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (\sqrt[7]{4^{7} \cdot (2 x^{5})} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.4.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (\sqrt[7]{4^{7} \cdot (2 x^{5})} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.5

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 {\sqrt[7]{8 x}}^{4} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.6

Rewrite ${\sqrt[7]{8 x}}^{4}$ as $\sqrt[7]{{(8 x)}^{4}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 \sqrt[7]{{(8 x)}^{4}} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.7

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 \sqrt[7]{8^{4} x^{4}} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.8

Raise $8$ to the power of $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 \sqrt[7]{4096 x^{4}} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.9

Rewrite $4096 x^{4}$ as $2^{7} \cdot (32 x^{4})$ .

Tap for more steps...

Step 5.2.4.8.9.1

Factor $128$ out of $4096$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 \sqrt[7]{128 (32) x^{4}} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.9.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 \sqrt[7]{2^{7} \cdot (32 x^{4})} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.9.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 \sqrt[7]{2^{7} \cdot (32 x^{4})} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.10

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 5 (2 \sqrt[7]{32 x^{4}}) \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.11

Multiply $2$ by $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 10 \sqrt[7]{32 x^{4}} \cdot 5 + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.12

Multiply $5$ by $10$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 {\sqrt[7]{8 x}}^{3} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.13

Rewrite ${\sqrt[7]{8 x}}^{3}$ as $\sqrt[7]{{(8 x)}^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 \sqrt[7]{{(8 x)}^{3}} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.14

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 \sqrt[7]{8^{3} x^{3}} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.15

Raise $8$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 \sqrt[7]{512 x^{3}} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.16

Rewrite $512 x^{3}$ as $2^{7} \cdot (4 x^{3})$ .

Tap for more steps...

Step 5.2.4.8.16.1

Factor $128$ out of $512$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 \sqrt[7]{128 (4) x^{3}} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.16.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 \sqrt[7]{2^{7} \cdot (4 x^{3})} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.16.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 \sqrt[7]{2^{7} \cdot (4 x^{3})} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.17

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 10 (2 \sqrt[7]{4 x^{3}}) \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.18

Multiply $2$ by $10$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 20 \sqrt[7]{4 x^{3}} \cdot 5^{2} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.19

Raise $5$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 20 \sqrt[7]{4 x^{3}} \cdot 25 + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.20

Multiply $25$ by $20$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 10 {\sqrt[7]{8 x}}^{2} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.21

Rewrite ${\sqrt[7]{8 x}}^{2}$ as $\sqrt[7]{{(8 x)}^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 10 \sqrt[7]{{(8 x)}^{2}} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.22

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 10 \sqrt[7]{8^{2} x^{2}} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.23

Raise $8$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 10 \sqrt[7]{64 x^{2}} \cdot 5^{3} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.24

Raise $5$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 10 \sqrt[7]{64 x^{2}} \cdot 125 + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.25

Multiply $125$ by $10$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 5 \sqrt[7]{8 x} \cdot 5^{4} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.26

Multiply $5$ by $5^{4}$ by adding the exponents.

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Step 5.2.4.8.26.1

Move $5^{4}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 5^{4} \cdot (5 \sqrt[7]{8 x}) + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.26.2

Multiply $5^{4}$ by $5$ .

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Step 5.2.4.8.26.2.1

Raise $5$ to the power of $1$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 5^{4} \cdot (5 \sqrt[7]{8 x}) + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.26.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 5^{4 + 1} \sqrt[7]{8 x} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.26.3

Add $4$ and $1$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 5^{5} \sqrt[7]{8 x} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.27

Raise $5$ to the power of $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 3125 \sqrt[7]{8 x} + 5^{5}) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.8.28

Raise $5$ to the power of $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}} + 50 \sqrt[7]{32 x^{4}} + 500 \sqrt[7]{4 x^{3}} + 1250 \sqrt[7]{64 x^{2}} + 3125 \sqrt[7]{8 x} + 3125) - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.9

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 525 (4 \sqrt[7]{2 x^{5}}) + 525 (50 \sqrt[7]{32 x^{4}}) + 525 (500 \sqrt[7]{4 x^{3}}) + 525 (1250 \sqrt[7]{64 x^{2}}) + 525 (3125 \sqrt[7]{8 x}) + 525 \cdot 3125 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.10

Simplify.

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Step 5.2.4.10.1

Multiply $4$ by $525$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 525 (50 \sqrt[7]{32 x^{4}}) + 525 (500 \sqrt[7]{4 x^{3}}) + 525 (1250 \sqrt[7]{64 x^{2}}) + 525 (3125 \sqrt[7]{8 x}) + 525 \cdot 3125 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.10.2

Multiply $50$ by $525$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 525 (500 \sqrt[7]{4 x^{3}}) + 525 (1250 \sqrt[7]{64 x^{2}}) + 525 (3125 \sqrt[7]{8 x}) + 525 \cdot 3125 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.10.3

Multiply $500$ by $525$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 525 (1250 \sqrt[7]{64 x^{2}}) + 525 (3125 \sqrt[7]{8 x}) + 525 \cdot 3125 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.10.4

Multiply $1250$ by $525$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 525 (3125 \sqrt[7]{8 x}) + 525 \cdot 3125 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.10.5

Multiply $3125$ by $525$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 525 \cdot 3125 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.10.6

Multiply $525$ by $3125$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 {(\sqrt[7]{8 x} + 5)}^{4} + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.11

Use the Binomial Theorem.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 ({\sqrt[7]{8 x}}^{4} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12

Simplify each term.

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Step 5.2.4.12.1

Rewrite ${\sqrt[7]{8 x}}^{4}$ as $\sqrt[7]{{(8 x)}^{4}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (\sqrt[7]{{(8 x)}^{4}} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.2

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (\sqrt[7]{8^{4} x^{4}} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.3

Raise $8$ to the power of $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (\sqrt[7]{4096 x^{4}} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.4

Rewrite $4096 x^{4}$ as $2^{7} \cdot (32 x^{4})$ .

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Step 5.2.4.12.4.1

Factor $128$ out of $4096$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (\sqrt[7]{128 (32) x^{4}} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.4.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (\sqrt[7]{2^{7} \cdot (32 x^{4})} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.4.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (\sqrt[7]{2^{7} \cdot (32 x^{4})} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.5

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 {\sqrt[7]{8 x}}^{3} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.6

Rewrite ${\sqrt[7]{8 x}}^{3}$ as $\sqrt[7]{{(8 x)}^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 \sqrt[7]{{(8 x)}^{3}} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.7

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 \sqrt[7]{8^{3} x^{3}} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.8

Raise $8$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 \sqrt[7]{512 x^{3}} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.9

Rewrite $512 x^{3}$ as $2^{7} \cdot (4 x^{3})$ .

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Step 5.2.4.12.9.1

Factor $128$ out of $512$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 \sqrt[7]{128 (4) x^{3}} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.9.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 \sqrt[7]{2^{7} \cdot (4 x^{3})} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.9.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 \sqrt[7]{2^{7} \cdot (4 x^{3})} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.10

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 4 (2 \sqrt[7]{4 x^{3}}) \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.11

Multiply $2$ by $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 8 \sqrt[7]{4 x^{3}} \cdot 5 + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.12

Multiply $5$ by $8$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 6 {\sqrt[7]{8 x}}^{2} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.13

Rewrite ${\sqrt[7]{8 x}}^{2}$ as $\sqrt[7]{{(8 x)}^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 6 \sqrt[7]{{(8 x)}^{2}} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.14

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 6 \sqrt[7]{8^{2} x^{2}} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.15

Raise $8$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 6 \sqrt[7]{64 x^{2}} \cdot 5^{2} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.16

Raise $5$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 6 \sqrt[7]{64 x^{2}} \cdot 25 + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.17

Multiply $25$ by $6$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 150 \sqrt[7]{64 x^{2}} + 4 \sqrt[7]{8 x} \cdot 5^{3} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.18

Raise $5$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 150 \sqrt[7]{64 x^{2}} + 4 \sqrt[7]{8 x} \cdot 125 + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.19

Multiply $125$ by $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 150 \sqrt[7]{64 x^{2}} + 500 \sqrt[7]{8 x} + 5^{4}) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.12.20

Raise $5$ to the power of $4$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}} + 40 \sqrt[7]{4 x^{3}} + 150 \sqrt[7]{64 x^{2}} + 500 \sqrt[7]{8 x} + 625) + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.13

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 4375 (2 \sqrt[7]{32 x^{4}}) - 4375 (40 \sqrt[7]{4 x^{3}}) - 4375 (150 \sqrt[7]{64 x^{2}}) - 4375 (500 \sqrt[7]{8 x}) - 4375 \cdot 625 + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.14

Simplify.

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Step 5.2.4.14.1

Multiply $2$ by $- 4375$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 4375 (40 \sqrt[7]{4 x^{3}}) - 4375 (150 \sqrt[7]{64 x^{2}}) - 4375 (500 \sqrt[7]{8 x}) - 4375 \cdot 625 + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.14.2

Multiply $40$ by $- 4375$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 4375 (150 \sqrt[7]{64 x^{2}}) - 4375 (500 \sqrt[7]{8 x}) - 4375 \cdot 625 + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.14.3

Multiply $150$ by $- 4375$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 4375 (500 \sqrt[7]{8 x}) - 4375 \cdot 625 + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.14.4

Multiply $500$ by $- 4375$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 4375 \cdot 625 + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.14.5

Multiply $- 4375$ by $625$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 {(\sqrt[7]{8 x} + 5)}^{3} - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.15

Use the Binomial Theorem.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 ({\sqrt[7]{8 x}}^{3} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16

Simplify each term.

Tap for more steps...

Step 5.2.4.16.1

Rewrite ${\sqrt[7]{8 x}}^{3}$ as $\sqrt[7]{{(8 x)}^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (\sqrt[7]{{(8 x)}^{3}} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.2

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (\sqrt[7]{8^{3} x^{3}} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.3

Raise $8$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (\sqrt[7]{512 x^{3}} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.4

Rewrite $512 x^{3}$ as $2^{7} \cdot (4 x^{3})$ .

Tap for more steps...

Step 5.2.4.16.4.1

Factor $128$ out of $512$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (\sqrt[7]{128 (4) x^{3}} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.4.2

Rewrite $128$ as $2^{7}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (\sqrt[7]{2^{7} \cdot (4 x^{3})} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.4.3

Add parentheses.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (\sqrt[7]{2^{7} \cdot (4 x^{3})} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.5

Pull terms out from under the radical.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 3 {\sqrt[7]{8 x}}^{2} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.6

Rewrite ${\sqrt[7]{8 x}}^{2}$ as $\sqrt[7]{{(8 x)}^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 3 \sqrt[7]{{(8 x)}^{2}} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.7

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 3 \sqrt[7]{8^{2} x^{2}} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.8

Raise $8$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 3 \sqrt[7]{64 x^{2}} \cdot 5 + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.9

Multiply $5$ by $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 15 \sqrt[7]{64 x^{2}} + 3 \sqrt[7]{8 x} \cdot 5^{2} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.10

Raise $5$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 15 \sqrt[7]{64 x^{2}} + 3 \sqrt[7]{8 x} \cdot 25 + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.11

Multiply $25$ by $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 15 \sqrt[7]{64 x^{2}} + 75 \sqrt[7]{8 x} + 5^{3}) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.16.12

Raise $5$ to the power of $3$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}} + 15 \sqrt[7]{64 x^{2}} + 75 \sqrt[7]{8 x} + 125) - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.17

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 21875 (2 \sqrt[7]{4 x^{3}}) + 21875 (15 \sqrt[7]{64 x^{2}}) + 21875 (75 \sqrt[7]{8 x}) + 21875 \cdot 125 - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.18

Simplify.

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Step 5.2.4.18.1

Multiply $2$ by $21875$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 21875 (15 \sqrt[7]{64 x^{2}}) + 21875 (75 \sqrt[7]{8 x}) + 21875 \cdot 125 - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.18.2

Multiply $15$ by $21875$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 21875 (75 \sqrt[7]{8 x}) + 21875 \cdot 125 - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.18.3

Multiply $75$ by $21875$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 21875 \cdot 125 - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.18.4

Multiply $21875$ by $125$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 {(\sqrt[7]{8 x} + 5)}^{2} + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.19

Rewrite ${(\sqrt[7]{8 x} + 5)}^{2}$ as $(\sqrt[7]{8 x} + 5) (\sqrt[7]{8 x} + 5)$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 ((\sqrt[7]{8 x} + 5) (\sqrt[7]{8 x} + 5)) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.20

Expand $(\sqrt[7]{8 x} + 5) (\sqrt[7]{8 x} + 5)$ using the FOIL Method.

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Step 5.2.4.20.1

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{8 x} (\sqrt[7]{8 x} + 5) + 5 (\sqrt[7]{8 x} + 5)) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.20.2

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{8 x} \sqrt[7]{8 x} + \sqrt[7]{8 x} \cdot 5 + 5 (\sqrt[7]{8 x} + 5)) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.20.3

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{8 x} \sqrt[7]{8 x} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21

Simplify and combine like terms.

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Step 5.2.4.21.1

Simplify each term.

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Step 5.2.4.21.1.1

Multiply $\sqrt[7]{8 x} \sqrt[7]{8 x}$ .

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Step 5.2.4.21.1.1.1

Raise $\sqrt[7]{8 x}$ to the power of $1$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{8 x} \sqrt[7]{8 x} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.1.2

Raise $\sqrt[7]{8 x}$ to the power of $1$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{8 x} \sqrt[7]{8 x} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 ({\sqrt[7]{8 x}}^{1 + 1} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 ({\sqrt[7]{8 x}}^{2} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.2

Rewrite ${\sqrt[7]{8 x}}^{2}$ as $\sqrt[7]{{(8 x)}^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{{(8 x)}^{2}} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.3

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{8^{2} x^{2}} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.4

Raise $8$ to the power of $2$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{64 x^{2}} + \sqrt[7]{8 x} \cdot 5 + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.5

Move $5$ to the left of $\sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{64 x^{2}} + 5 \cdot \sqrt[7]{8 x} + 5 \sqrt[7]{8 x} + 5 \cdot 5) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.1.6

Multiply $5$ by $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{64 x^{2}} + 5 \sqrt[7]{8 x} + 5 \sqrt[7]{8 x} + 25) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.21.2

Add $5 \sqrt[7]{8 x}$ and $5 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 (\sqrt[7]{64 x^{2}} + 10 \sqrt[7]{8 x} + 25) + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.22

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 65625 (10 \sqrt[7]{8 x}) - 65625 \cdot 25 + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.23

Simplify.

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Step 5.2.4.23.1

Multiply $10$ by $- 65625$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 65625 \cdot 25 + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.23.2

Multiply $- 65625$ by $25$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 (\sqrt[7]{8 x} + 5) - 78125}{8}$

Step 5.2.4.24

Apply the distributive property.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 109375 \cdot 5 - 78125}{8}$

Step 5.2.4.25

Multiply $109375$ by $5$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5

Simplify terms.

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Step 5.2.5.1

Combine the opposite terms in $8 x + 140 \sqrt[7]{16 x^{6}} + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 140 \sqrt[7]{16 x^{6}} - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125$ .

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Step 5.2.5.1.1

Subtract $140 \sqrt[7]{16 x^{6}}$ from $140 \sqrt[7]{16 x^{6}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 0 + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.2

Add $8 x$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 26250 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 26250 \sqrt[7]{32 x^{4}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.3

Add $- 26250 \sqrt[7]{32 x^{4}}$ and $26250 \sqrt[7]{32 x^{4}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} + 0 - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.4

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 8750 \sqrt[7]{32 x^{4}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 8750 \sqrt[7]{32 x^{4}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.5

Subtract $8750 \sqrt[7]{32 x^{4}}$ from $8750 \sqrt[7]{32 x^{4}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 0 + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.6

Add $8 x + 2100 \sqrt[7]{2 x^{5}}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 656250 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{64 x^{2}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.7

Subtract $656250 \sqrt[7]{64 x^{2}}$ from $656250 \sqrt[7]{64 x^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 0 + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.8

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 328125 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 328125 \sqrt[7]{64 x^{2}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.9

Add $- 328125 \sqrt[7]{64 x^{2}}$ and $328125 \sqrt[7]{64 x^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} + 0 - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.10

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} - 2734375 + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 2734375 - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.11

Add $- 2734375$ and $2734375$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 0 + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.12

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 65625 \sqrt[7]{64 x^{2}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 65625 \sqrt[7]{64 x^{2}} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.13

Subtract $65625 \sqrt[7]{64 x^{2}}$ from $65625 \sqrt[7]{64 x^{2}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 0 + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.14

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 1640625 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} - 1640625 + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.15

Subtract $1640625$ from $1640625$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} + 0 - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.16

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} - 546875 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x} + 546875 - 78125}{8}$

Step 5.2.5.1.17

Add $- 546875$ and $546875$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 0 + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x} - 78125}{8}$

Step 5.2.5.1.18

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 78125 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x} - 78125}{8}$

Step 5.2.5.1.19

Subtract $78125$ from $78125$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} + 0 - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.1.20

Add $8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x}$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 4200 \sqrt[7]{2 x^{5}} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.2

Subtract $4200 \sqrt[7]{2 x^{5}}$ from $2100 \sqrt[7]{2 x^{5}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x - 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.3

Combine the opposite terms in $8 x - 2100 \sqrt[7]{2 x^{5}} + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 2100 \sqrt[7]{2 x^{5}} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}$ .

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Step 5.2.5.3.1

Add $- 2100 \sqrt[7]{2 x^{5}}$ and $2100 \sqrt[7]{2 x^{5}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 0 + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.3.2

Add $8 x$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 656250 \sqrt[7]{8 x} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.4

Subtract $175000 \sqrt[7]{4 x^{3}}$ from $43750 \sqrt[7]{4 x^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x - 131250 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 262500 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.5

Add $- 131250 \sqrt[7]{4 x^{3}}$ and $262500 \sqrt[7]{4 x^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 131250 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 175000 \sqrt[7]{4 x^{3}} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.6

Subtract $175000 \sqrt[7]{4 x^{3}}$ from $131250 \sqrt[7]{4 x^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x - 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.7

Combine the opposite terms in $8 x - 43750 \sqrt[7]{4 x^{3}} + 109375 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 2187500 \sqrt[7]{8 x} + 43750 \sqrt[7]{4 x^{3}} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}$ .

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Step 5.2.5.7.1

Add $- 43750 \sqrt[7]{4 x^{3}}$ and $43750 \sqrt[7]{4 x^{3}}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 0 + 109375 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 2187500 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.7.2

Add $8 x$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 109375 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 2187500 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.8

Subtract $656250 \sqrt[7]{8 x}$ from $109375 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x - 546875 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 2187500 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.9

Add $- 546875 \sqrt[7]{8 x}$ and $1640625 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 1093750 \sqrt[7]{8 x} - 2187500 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.10

Subtract $2187500 \sqrt[7]{8 x}$ from $1093750 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x - 1093750 \sqrt[7]{8 x} + 1640625 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.11

Add $- 1093750 \sqrt[7]{8 x}$ and $1640625 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 546875 \sqrt[7]{8 x} - 656250 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.12

Subtract $656250 \sqrt[7]{8 x}$ from $546875 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x - 109375 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}}{8}$

Step 5.2.5.13

Combine the opposite terms in $8 x - 109375 \sqrt[7]{8 x} + 109375 \sqrt[7]{8 x}$ .

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Step 5.2.5.13.1

Add $- 109375 \sqrt[7]{8 x}$ and $109375 \sqrt[7]{8 x}$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x + 0}{8}$

Step 5.2.5.13.2

Add $8 x$ and $0$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x}{8}$

Step 5.2.5.14

Cancel the common factor of $8$ .

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Step 5.2.5.14.1

Cancel the common factor.

$f^{- 1} (\sqrt[7]{8 x} + 5) = \frac{8 x}{8}$

Step 5.2.5.14.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt[7]{8 x} + 5) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}) = \sqrt[7]{8 (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8})} + 5$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}) = \sqrt[7]{8 (\frac{x^{7} - 35 x^{6} + 525 x^{5} - 4375 x^{4} + 21875 x^{3} - 65625 x^{2} + 109375 x - 78125}{8})} + 5$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

Make each term match the terms from the binomial theorem formula.

$f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}) = \sqrt[7]{8 (\frac{x^{7} - 7 \cdot (5 x^{6}) + 21 \cdot (5^{2} x^{5}) - 35 \cdot (5^{3} x^{4}) + 35 \cdot (5^{4} x^{3}) - 21 \cdot (5^{5} x^{2}) + 7 \cdot (5^{6} x) - 1 \cdot 5^{7}}{8})} + 5$

Step 5.3.3.2.2

Factor $x^{7} - 7 \cdot 5 x^{6} + 21 \cdot 5^{2} x^{5} - 35 \cdot 5^{3} x^{4} + 35 \cdot 5^{4} x^{3} - 21 \cdot 5^{5} x^{2} + 7 \cdot 5^{6} x - 1 \cdot 5^{7}$ using the binomial theorem.

Step 5.3.3.3

Combine $8$ and $\frac{{(x - 5)}^{7}}{8}$ .

Step 5.3.3.4

Reduce the expression by cancelling the common factors.

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Step 5.3.3.4.1

Reduce the expression $\frac{8 {(x - 5)}^{7}}{8}$ by cancelling the common factors.

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Step 5.3.3.4.1.1

Cancel the common factor.

Step 5.3.3.4.1.2

Rewrite the expression.

Step 5.3.3.4.2

Divide ${(x - 5)}^{7}$ by $1$ .

Step 5.3.3.5

Pull terms out from under the radical, assuming real numbers.

$f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}) = x - 5 + 5$

Step 5.3.4

Combine the opposite terms in $x - 5 + 5$ .

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Step 5.3.4.1

Add $- 5$ and $5$ .

$f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}$ is the inverse of $f (x) = \sqrt[7]{8 x} + 5$ .

$f^{- 1} (x) = \frac{x^{7}}{8} - \frac{35 x^{6}}{8} + \frac{525 x^{5}}{8} - \frac{4375 x^{4}}{8} + \frac{21875 x^{3}}{8} - \frac{65625 x^{2}}{8} + \frac{109375 x}{8} - \frac{78125}{8}$