Algebra Examples

$\sqrt[3]{5 x - 10} - 8$

Step 1

Interchange the variables.

$x = \sqrt[3]{5 y - 10} - 8$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\sqrt[3]{5 y - 10} - 8 = x$ .

$\sqrt[3]{5 y - 10} - 8 = x$

Step 2.2

Add $8$ to both sides of the equation.

$\sqrt[3]{5 y - 10} = x + 8$

Step 2.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{5 y - 10}}^{3} = {(x + 8)}^{3}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 y - 10}$ as ${(5 y - 10)}^{\frac{1}{3}}$ .

${({(5 y - 10)}^{\frac{1}{3}})}^{3} = {(x + 8)}^{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${({(5 y - 10)}^{\frac{1}{3}})}^{3}$ .

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Step 2.4.2.1.1

Multiply the exponents in ${({(5 y - 10)}^{\frac{1}{3}})}^{3}$ .

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Step 2.4.2.1.1.1

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

${(5 y - 10)}^{\frac{1}{3} \cdot 3} = {(x + 8)}^{3}$

Step 2.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.4.2.1.1.2.1

Cancel the common factor.

${(5 y - 10)}^{\frac{1}{3} \cdot 3} = {(x + 8)}^{3}$

Step 2.4.2.1.1.2.2

Rewrite the expression.

${(5 y - 10)}^{1} = {(x + 8)}^{3}$

Step 2.4.2.1.2

Simplify.

$5 y - 10 = {(x + 8)}^{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(x + 8)}^{3}$ .

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Step 2.4.3.1.1

Use the Binomial Theorem.

$5 y - 10 = x^{3} + 3 x^{2} \cdot 8 + 3 x \cdot 8^{2} + 8^{3}$

Step 2.4.3.1.2

Simplify each term.

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Step 2.4.3.1.2.1

Multiply $8$ by $3$ .

$5 y - 10 = x^{3} + 24 x^{2} + 3 x \cdot 8^{2} + 8^{3}$

Step 2.4.3.1.2.2

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

$5 y - 10 = x^{3} + 24 x^{2} + 3 x \cdot 64 + 8^{3}$

Step 2.4.3.1.2.3

Multiply $64$ by $3$ .

$5 y - 10 = x^{3} + 24 x^{2} + 192 x + 8^{3}$

Step 2.4.3.1.2.4

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

$5 y - 10 = x^{3} + 24 x^{2} + 192 x + 512$

Step 2.5

Solve for $y$ .

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Step 2.5.1

Move all terms not containing $y$ to the right side of the equation.

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Step 2.5.1.1

Add $10$ to both sides of the equation.

$5 y = x^{3} + 24 x^{2} + 192 x + 512 + 10$

Step 2.5.1.2

Add $512$ and $10$ .

$5 y = x^{3} + 24 x^{2} + 192 x + 522$

Step 2.5.2

Divide each term in $5 y = x^{3} + 24 x^{2} + 192 x + 522$ by $5$ and simplify.

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Step 2.5.2.1

Divide each term in $5 y = x^{3} + 24 x^{2} + 192 x + 522$ by $5$ .

$\frac{5 y}{5} = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$

Step 2.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$

Step 3

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

$f^{- 1} (x) = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$

Step 4

Verify if $f^{- 1} (x) = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$ is the inverse of $f (x) = \sqrt[3]{5 x - 10} - 8$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (\sqrt[3]{5 x - 10} - 8)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(\sqrt[3]{5 x - 10} - 8)}^{3}}{5} + \frac{24 {(\sqrt[3]{5 x - 10} - 8)}^{2}}{5} + \frac{192 (\sqrt[3]{5 x - 10} - 8)}{5} + \frac{522}{5}$

Step 4.2.3

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(\sqrt[3]{5 x - 10} - 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

Factor $5$ out of $5 x - 10$ .

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Step 4.2.4.1.1

Factor $5$ out of $5 x$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(\sqrt[3]{5 (x) - 10} - 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.1.2

Factor $5$ out of $- 10$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(\sqrt[3]{5 x + 5 \cdot - 2} - 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.1.3

Factor $5$ out of $5 x + 5 \cdot - 2$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(\sqrt[3]{5 (x - 2)} - 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.2

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{\sqrt[3]{5 (x - 2)}}^{3} + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3

Simplify each term.

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Step 4.2.4.3.1

Rewrite ${\sqrt[3]{5 (x - 2)}}^{3}$ as $5 (x - 2)$ .

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Step 4.2.4.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5 (x - 2)}$ as ${(5 (x - 2))}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{({(5 (x - 2))}^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.1.2

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

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(5 (x - 2))}^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.1.3

Combine $\frac{1}{3}$ and $3$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{{(5 (x - 2))}^{\frac{3}{3}} + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.1.4

Cancel the common factor of $3$ .

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Step 4.2.4.3.1.4.1

Cancel the common factor.

Step 4.2.4.3.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{(5 (x - 2)) + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.1.5

Simplify.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 (x - 2) + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x + 5 \cdot - 2 + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.3

Multiply $5$ by $- 2$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 + 3 {\sqrt[3]{5 (x - 2)}}^{2} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.4

Rewrite ${\sqrt[3]{5 (x - 2)}}^{2}$ as $\sqrt[3]{{(5 (x - 2))}^{2}}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 + 3 \sqrt[3]{{(5 (x - 2))}^{2}} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.5

Apply the product rule to $5 (x - 2)$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 + 3 \sqrt[3]{5^{2} {(x - 2)}^{2}} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.6

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

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 + 3 \sqrt[3]{25 {(x - 2)}^{2}} \cdot - 8 + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.7

Multiply $- 8$ by $3$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 3 \sqrt[3]{5 (x - 2)} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.8

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

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 3 \sqrt[3]{5 (x - 2)} \cdot 64 + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.9

Multiply $64$ by $3$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + {(- 8)}^{3} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.3.10

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

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 10 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} - 512 + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.4

Subtract $512$ from $- 10$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 {(\sqrt[3]{5 x - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.5

Factor $5$ out of $5 x - 10$ .

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Step 4.2.4.5.1

Factor $5$ out of $5 x$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 {(\sqrt[3]{5 (x) - 10} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.5.2

Factor $5$ out of $- 10$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 {(\sqrt[3]{5 x + 5 \cdot - 2} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.5.3

Factor $5$ out of $5 x + 5 \cdot - 2$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 {(\sqrt[3]{5 (x - 2)} - 8)}^{2} + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.6

Rewrite ${(\sqrt[3]{5 (x - 2)} - 8)}^{2}$ as $(\sqrt[3]{5 (x - 2)} - 8) (\sqrt[3]{5 (x - 2)} - 8)$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 ((\sqrt[3]{5 (x - 2)} - 8) (\sqrt[3]{5 (x - 2)} - 8)) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.7

Expand $(\sqrt[3]{5 (x - 2)} - 8) (\sqrt[3]{5 (x - 2)} - 8)$ using the FOIL Method.

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Step 4.2.4.7.1

Apply the distributive property.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{5 (x - 2)} (\sqrt[3]{5 (x - 2)} - 8) - 8 (\sqrt[3]{5 (x - 2)} - 8)) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.7.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{5 (x - 2)} \sqrt[3]{5 (x - 2)} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 (\sqrt[3]{5 (x - 2)} - 8)) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.7.3

Apply the distributive property.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{5 (x - 2)} \sqrt[3]{5 (x - 2)} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8

Simplify and combine like terms.

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Step 4.2.4.8.1

Simplify each term.

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Step 4.2.4.8.1.1

Multiply $\sqrt[3]{5 (x - 2)} \sqrt[3]{5 (x - 2)}$ .

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Step 4.2.4.8.1.1.1

Raise $\sqrt[3]{5 (x - 2)}$ to the power of $1$ .

Step 4.2.4.8.1.1.2

Raise $\sqrt[3]{5 (x - 2)}$ to the power of $1$ .

Step 4.2.4.8.1.1.3

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

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 ({\sqrt[3]{5 (x - 2)}}^{1 + 1} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 ({\sqrt[3]{5 (x - 2)}}^{2} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.1.2

Rewrite ${\sqrt[3]{5 (x - 2)}}^{2}$ as $\sqrt[3]{{(5 (x - 2))}^{2}}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{{(5 (x - 2))}^{2}} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.1.3

Apply the product rule to $5 (x - 2)$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{5^{2} {(x - 2)}^{2}} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.1.4

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

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{25 {(x - 2)}^{2}} + \sqrt[3]{5 (x - 2)} \cdot - 8 - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.1.5

Move $- 8$ to the left of $\sqrt[3]{5 (x - 2)}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{25 {(x - 2)}^{2}} - 8 \cdot \sqrt[3]{5 (x - 2)} - 8 \sqrt[3]{5 (x - 2)} - 8 \cdot - 8) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.1.6

Multiply $- 8$ by $- 8$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{25 {(x - 2)}^{2}} - 8 \sqrt[3]{5 (x - 2)} - 8 \sqrt[3]{5 (x - 2)} + 64) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.8.2

Subtract $8 \sqrt[3]{5 (x - 2)}$ from $- 8 \sqrt[3]{5 (x - 2)}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 (\sqrt[3]{25 {(x - 2)}^{2}} - 16 \sqrt[3]{5 (x - 2)} + 64) + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.9

Apply the distributive property.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 \sqrt[3]{25 {(x - 2)}^{2}} + 24 (- 16 \sqrt[3]{5 (x - 2)}) + 24 \cdot 64 + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.10

Simplify.

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Step 4.2.4.10.1

Multiply $- 16$ by $24$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 \sqrt[3]{25 {(x - 2)}^{2}} - 384 \sqrt[3]{5 (x - 2)} + 24 \cdot 64 + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.10.2

Multiply $24$ by $64$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 \sqrt[3]{25 {(x - 2)}^{2}} - 384 \sqrt[3]{5 (x - 2)} + 1536 + 192 (\sqrt[3]{5 x - 10} - 8) + 522}{5}$

Step 4.2.4.11

Factor $5$ out of $5 x - 10$ .

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Step 4.2.4.11.1

Factor $5$ out of $5 x$ .

Step 4.2.4.11.2

Factor $5$ out of $- 10$ .

Step 4.2.4.11.3

Factor $5$ out of $5 x + 5 \cdot - 2$ .

Step 4.2.4.12

Apply the distributive property.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 \sqrt[3]{25 {(x - 2)}^{2}} - 384 \sqrt[3]{5 (x - 2)} + 1536 + 192 \sqrt[3]{5 (x - 2)} + 192 \cdot - 8 + 522}{5}$

Step 4.2.4.13

Multiply $192$ by $- 8$ .

Step 4.2.5

Simplify terms.

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Step 4.2.5.1

Combine the opposite terms in $5 x - 522 - 24 \sqrt[3]{25 {(x - 2)}^{2}} + 192 \sqrt[3]{5 (x - 2)} + 24 \sqrt[3]{25 {(x - 2)}^{2}} - 384 \sqrt[3]{5 (x - 2)} + 1536 + 192 \sqrt[3]{5 (x - 2)} - 1536 + 522$ .

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Step 4.2.5.1.1

Add $- 24 \sqrt[3]{25 {(x - 2)}^{2}}$ and $24 \sqrt[3]{25 {(x - 2)}^{2}}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 + 0 + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)} + 1536 + 192 \sqrt[3]{5 (x - 2)} - 1536 + 522}{5}$

Step 4.2.5.1.2

Add $5 x - 522$ and $0$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)} + 1536 + 192 \sqrt[3]{5 (x - 2)} - 1536 + 522}{5}$

Step 4.2.5.1.3

Subtract $1536$ from $1536$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)} + 0 + 192 \sqrt[3]{5 (x - 2)} + 522}{5}$

Step 4.2.5.1.4

Add $5 x - 522 + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)}$ and $0$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 522 + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)} + 192 \sqrt[3]{5 (x - 2)} + 522}{5}$

Step 4.2.5.1.5

Add $- 522$ and $522$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x + 0 + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)} + 192 \sqrt[3]{5 (x - 2)}}{5}$

Step 4.2.5.1.6

Add $5 x$ and $0$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x + 192 \sqrt[3]{5 (x - 2)} - 384 \sqrt[3]{5 (x - 2)} + 192 \sqrt[3]{5 (x - 2)}}{5}$

Step 4.2.5.2

Subtract $384 \sqrt[3]{5 (x - 2)}$ from $192 \sqrt[3]{5 (x - 2)}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x - 192 \sqrt[3]{5 (x - 2)} + 192 \sqrt[3]{5 (x - 2)}}{5}$

Step 4.2.5.3

Combine the opposite terms in $5 x - 192 \sqrt[3]{5 (x - 2)} + 192 \sqrt[3]{5 (x - 2)}$ .

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Step 4.2.5.3.1

Add $- 192 \sqrt[3]{5 (x - 2)}$ and $192 \sqrt[3]{5 (x - 2)}$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x + 0}{5}$

Step 4.2.5.3.2

Add $5 x$ and $0$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x}{5}$

Step 4.2.5.4

Cancel the common factor of $5$ .

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Step 4.2.5.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = \frac{5 x}{5}$

Step 4.2.5.4.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt[3]{5 x - 10} - 8) = x$

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) - 10} - 8$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Factor $5$ out of $5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) - 10$ .

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Step 4.3.3.1.1

Factor $5$ out of $- 10$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) + 5 \cdot - 2} - 8$

Step 4.3.3.1.2

Factor $5$ out of $5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) + 5 \cdot - 2$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5} - 2)} - 8$

Step 4.3.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5} - 2 \cdot \frac{5}{5})} - 8$

Step 4.3.3.3

Combine $- 2$ and $\frac{5}{5}$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5} + \frac{- 2 \cdot 5}{5})} - 8$

Step 4.3.3.4

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522 - 2 \cdot 5}{5})} - 8$

Step 4.3.3.5

Simplify the numerator.

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Step 4.3.3.5.1

Multiply $- 2$ by $5$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522 - 10}{5})} - 8$

Step 4.3.3.5.2

Subtract $10$ from $522$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{512}{5})} - 8$

Step 4.3.3.6

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{5 (\frac{x^{3} + 24 x^{2} + 192 x + 512}{5})} - 8$

Step 4.3.3.7

Combine $5$ and $\frac{x^{3} + 24 x^{2} + 192 x + 512}{5}$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{\frac{5 (x^{3} + 24 x^{2} + 192 x + 512)}{5}} - 8$

Step 4.3.3.8

Reduce the expression by cancelling the common factors.

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Step 4.3.3.8.1

Reduce the expression $\frac{5 (x^{3} + 24 x^{2} + 192 x + 512)}{5}$ by cancelling the common factors.

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Step 4.3.3.8.1.1

Cancel the common factor.

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{\frac{5 (x^{3} + 24 x^{2} + 192 x + 512)}{5}} - 8$

Step 4.3.3.8.1.2

Rewrite the expression.

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{\frac{x^{3} + 24 x^{2} + 192 x + 512}{1}} - 8$

Step 4.3.3.8.2

Divide $x^{3} + 24 x^{2} + 192 x + 512$ by $1$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{x^{3} + 24 x^{2} + 192 x + 512} - 8$

Step 4.3.3.9

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

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{x^{3} + 3 \cdot (8 x^{2}) + 3 \cdot (8^{2} x) + 8^{3}} - 8$

Step 4.3.3.10

Factor $x^{3} + 3 \cdot 8 x^{2} + 3 \cdot 8^{2} x + 8^{3}$ using the binomial theorem.

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = \sqrt[3]{{(x + 8)}^{3}} - 8$

Step 4.3.3.11

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

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = x + 8 - 8$

Step 4.3.4

Combine the opposite terms in $x + 8 - 8$ .

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Step 4.3.4.1

Subtract $8$ from $8$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (\frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$ is the inverse of $f (x) = \sqrt[3]{5 x - 10} - 8$ .

$f^{- 1} (x) = \frac{x^{3}}{5} + \frac{24 x^{2}}{5} + \frac{192 x}{5} + \frac{522}{5}$