Algebra Examples

$f (x) = 5 x^{\frac{1}{5}} + 3$

Step 1

Write $f (x) = 5 x^{\frac{1}{5}} + 3$ as an equation.

$y = 5 x^{\frac{1}{5}} + 3$

Step 2

Interchange the variables.

$x = 5 y^{\frac{1}{5}} + 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 y^{\frac{1}{5}} + 3 = x$ .

$5 y^{\frac{1}{5}} + 3 = x$

Step 3.2

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

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Step 3.2.1

Subtract $3$ from both sides of the equation.

$5 y^{\frac{1}{5}} - x = - 3$

Step 3.2.2

Add $x$ to both sides of the equation.

$5 y^{\frac{1}{5}} = - 3 + x$

Step 3.3

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

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

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

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

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Step 3.4.1.1.1

Apply the product rule to $5 y^{\frac{1}{5}}$ .

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

Step 3.4.1.1.2

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

$3125 {(y^{\frac{1}{5}})}^{5} = {(- 3 + x)}^{5}$

Step 3.4.1.1.3

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

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Step 3.4.1.1.3.1

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

$3125 y^{\frac{1}{5} \cdot 5} = {(- 3 + x)}^{5}$

Step 3.4.1.1.3.2

Cancel the common factor of $5$ .

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Step 3.4.1.1.3.2.1

Cancel the common factor.

$3125 y^{\frac{1}{5} \cdot 5} = {(- 3 + x)}^{5}$

Step 3.4.1.1.3.2.2

Rewrite the expression.

$3125 y^{1} = {(- 3 + x)}^{5}$

Step 3.4.1.1.4

Simplify.

$3125 y = {(- 3 + x)}^{5}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(- 3 + x)}^{5}$ .

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Step 3.4.2.1.1

Use the Binomial Theorem.

$3125 y = {(- 3)}^{5} + 5 {(- 3)}^{4} x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2

Simplify each term.

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Step 3.4.2.1.2.1

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

$3125 y = - 243 + 5 {(- 3)}^{4} x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.2

Raise $- 3$ to the power of $4$ .

$3125 y = - 243 + 5 \cdot 81 x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.3

Multiply $5$ by $81$ .

$3125 y = - 243 + 405 x + 10 {(- 3)}^{3} x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.4

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

$3125 y = - 243 + 405 x + 10 \cdot - 27 x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.5

Multiply $10$ by $- 27$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 10 {(- 3)}^{2} x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.6

Raise $- 3$ to the power of $2$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 10 \cdot 9 x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.7

Multiply $10$ by $9$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} + 5 \cdot - 3 x^{4} + x^{5}$

Step 3.4.2.1.2.8

Multiply $5$ by $- 3$ .

$3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} - 15 x^{4} + x^{5}$

Step 3.5

Divide each term in $3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} - 15 x^{4} + x^{5}$ by $3125$ and simplify.

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Step 3.5.1

Divide each term in $3125 y = - 243 + 405 x - 270 x^{2} + 90 x^{3} - 15 x^{4} + x^{5}$ by $3125$ .

$\frac{3125 y}{3125} = \frac{- 243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Cancel the common factor of $3125$ .

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Step 3.5.2.1.1

Cancel the common factor.

$\frac{3125 y}{3125} = \frac{- 243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

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.

$y = - \frac{243}{3125} + \frac{405 x}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.2

Cancel the common factor of $405$ and $3125$ .

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Step 3.5.3.1.2.1

Factor $5$ out of $405 x$ .

$y = - \frac{243}{3125} + \frac{5 (81 x)}{3125} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.2.2

Cancel the common factors.

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Step 3.5.3.1.2.2.1

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{5 (81 x)}{5 (625)} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.2.2.2

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{5 (81 x)}{5 \cdot 625} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.2.2.3

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{- 270 x^{2}}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.3

Cancel the common factor of $- 270$ and $3125$ .

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Step 3.5.3.1.3.1

Factor $5$ out of $- 270 x^{2}$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{5 (- 54 x^{2})}{3125} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.3.2

Cancel the common factors.

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Step 3.5.3.1.3.2.1

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{5 (- 54 x^{2})}{5 (625)} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.3.2.2

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{5 (- 54 x^{2})}{5 \cdot 625} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.3.2.3

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} + \frac{- 54 x^{2}}{625} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.4

Move the negative in front of the fraction.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{90 x^{3}}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.5

Cancel the common factor of $90$ and $3125$ .

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Step 3.5.3.1.5.1

Factor $5$ out of $90 x^{3}$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{5 (18 x^{3})}{3125} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.5.2

Cancel the common factors.

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Step 3.5.3.1.5.2.1

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{5 (18 x^{3})}{5 (625)} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.5.2.2

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{5 (18 x^{3})}{5 \cdot 625} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.5.2.3

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{- 15 x^{4}}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.6

Cancel the common factor of $- 15$ and $3125$ .

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Step 3.5.3.1.6.1

Factor $5$ out of $- 15 x^{4}$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{5 (- 3 x^{4})}{3125} + \frac{x^{5}}{3125}$

Step 3.5.3.1.6.2

Cancel the common factors.

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Step 3.5.3.1.6.2.1

Factor $5$ out of $3125$ .

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{5 (- 3 x^{4})}{5 (625)} + \frac{x^{5}}{3125}$

Step 3.5.3.1.6.2.2

Cancel the common factor.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{5 (- 3 x^{4})}{5 \cdot 625} + \frac{x^{5}}{3125}$

Step 3.5.3.1.6.2.3

Rewrite the expression.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} + \frac{- 3 x^{4}}{625} + \frac{x^{5}}{3125}$

Step 3.5.3.1.7

Move the negative in front of the fraction.

$y = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$

Step 4

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

$f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$

Step 5

Verify if $f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$ is the inverse of $f (x) = 5 x^{\frac{1}{5}} + 3$ .

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{81 (5 x^{\frac{1}{5}} + 3)}{625} - \frac{54 {(5 x^{\frac{1}{5}} + 3)}^{2}}{625} + \frac{18 {(5 x^{\frac{1}{5}} + 3)}^{3}}{625} - \frac{3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125}$

Step 5.2.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{81 (5 x^{\frac{1}{5}} + 3) - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{81 (5 x^{\frac{1}{5}}) + 81 \cdot 3 - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.2

Multiply $5$ by $81$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 81 \cdot 3 - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.3

Multiply $81$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 {(5 x^{\frac{1}{5}} + 3)}^{2} + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.4

Rewrite ${(5 x^{\frac{1}{5}} + 3)}^{2}$ as $(5 x^{\frac{1}{5}} + 3) (5 x^{\frac{1}{5}} + 3)$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 ((5 x^{\frac{1}{5}} + 3) (5 x^{\frac{1}{5}} + 3)) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.5

Expand $(5 x^{\frac{1}{5}} + 3) (5 x^{\frac{1}{5}} + 3)$ using the FOIL Method.

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Step 5.2.4.5.1

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}} + 3) + 3 (5 x^{\frac{1}{5}} + 3)) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.5.2

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}} + 3)) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.5.3

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6

Simplify and combine like terms.

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Step 5.2.4.6.1

Simplify each term.

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Step 5.2.4.6.1.1

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{1}{5}} x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.2

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 5.2.4.6.1.2.1

Move $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 (x^{\frac{1}{5}} x^{\frac{1}{5}})) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.2.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{1}{5} + \frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.2.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{1 + 1}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.2.4

Add $1$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (5 \cdot (5 x^{\frac{2}{5}}) + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.3

Multiply $5$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.4

Multiply $3$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 3 (5 x^{\frac{1}{5}}) + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.5

Multiply $5$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 15 x^{\frac{1}{5}} + 3 \cdot 3) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.1.6

Multiply $3$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 15 x^{\frac{1}{5}} + 9) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.6.2

Add $15 x^{\frac{1}{5}}$ and $15 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 9) + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.7

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 54 (25 x^{\frac{2}{5}}) - 54 (30 x^{\frac{1}{5}}) - 54 \cdot 9 + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.8

Simplify.

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Step 5.2.4.8.1

Multiply $25$ by $- 54$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 54 (30 x^{\frac{1}{5}}) - 54 \cdot 9 + 18 {(5 x^{\frac{1}{5}} + 3)}^{3} - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.8.2

Multiply $30$ by $- 54$ .

Step 5.2.4.8.3

Multiply $- 54$ by $9$ .

Step 5.2.4.9

Use the Binomial Theorem.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 ({(5 x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10

Simplify each term.

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Step 5.2.4.10.1

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (5^{3} {(x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 {(x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.3

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

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Step 5.2.4.10.3.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{1}{5} \cdot 3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.3.2

Combine $\frac{1}{5}$ and $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.4

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.5

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (25 {(x^{\frac{1}{5}})}^{2}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.6

Multiply the exponents in ${(x^{\frac{1}{5}})}^{2}$ .

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Step 5.2.4.10.6.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (25 x^{\frac{1}{5} \cdot 2}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.6.2

Combine $\frac{1}{5}$ and $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 3 (25 x^{\frac{2}{5}}) \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.7

Multiply $25$ by $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 75 x^{\frac{2}{5}} \cdot 3 + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.8

Multiply $3$ by $75$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3 (5 x^{\frac{1}{5}}) \cdot 3^{2} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.9

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 5.2.4.10.9.1

Move $3^{2}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{2} \cdot (3 (5 x^{\frac{1}{5}})) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.9.2

Multiply $3^{2}$ by $3$ .

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Step 5.2.4.10.9.2.1

Raise $3$ to the power of $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{2} \cdot (3 (5 x^{\frac{1}{5}})) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.9.2.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{2 + 1} (5 x^{\frac{1}{5}}) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.9.3

Add $2$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 3^{3} (5 x^{\frac{1}{5}}) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.10

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 27 (5 x^{\frac{1}{5}}) + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.11

Multiply $5$ by $27$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 135 x^{\frac{1}{5}} + 3^{3}) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.10.12

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}} + 225 x^{\frac{2}{5}} + 135 x^{\frac{1}{5}} + 27) - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.11

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 18 (125 x^{\frac{3}{5}}) + 18 (225 x^{\frac{2}{5}}) + 18 (135 x^{\frac{1}{5}}) + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.12

Simplify.

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Step 5.2.4.12.1

Multiply $125$ by $18$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 18 (225 x^{\frac{2}{5}}) + 18 (135 x^{\frac{1}{5}}) + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.12.2

Multiply $225$ by $18$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 18 (135 x^{\frac{1}{5}}) + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.12.3

Multiply $135$ by $18$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 18 \cdot 27 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.12.4

Multiply $18$ by $27$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 {(5 x^{\frac{1}{5}} + 3)}^{4}}{625}$

Step 5.2.4.13

Use the Binomial Theorem.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 ({(5 x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14

Simplify each term.

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Step 5.2.4.14.1

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (5^{4} {(x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 {(x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.3

Multiply the exponents in ${(x^{\frac{1}{5}})}^{4}$ .

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Step 5.2.4.14.3.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{1}{5} \cdot 4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.3.2

Combine $\frac{1}{5}$ and $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.4

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (5^{3} {(x^{\frac{1}{5}})}^{3}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.5

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (125 {(x^{\frac{1}{5}})}^{3}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.6

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

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Step 5.2.4.14.6.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (125 x^{\frac{1}{5} \cdot 3}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.6.2

Combine $\frac{1}{5}$ and $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 4 (125 x^{\frac{3}{5}}) \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.7

Multiply $125$ by $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 500 x^{\frac{3}{5}} \cdot 3 + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.8

Multiply $3$ by $500$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.9

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.10

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (25 {(x^{\frac{1}{5}})}^{2}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.11

Multiply the exponents in ${(x^{\frac{1}{5}})}^{2}$ .

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Step 5.2.4.14.11.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (25 x^{\frac{1}{5} \cdot 2}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.11.2

Combine $\frac{1}{5}$ and $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 6 (25 x^{\frac{2}{5}}) \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.12

Multiply $25$ by $6$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} \cdot 3^{2} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.13

Raise $3$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} \cdot 9 + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.14

Multiply $9$ by $150$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 4 (5 x^{\frac{1}{5}}) \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.15

Multiply $5$ by $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 20 x^{\frac{1}{5}} \cdot 3^{3} + 3^{4})}{625}$

Step 5.2.4.14.16

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 20 x^{\frac{1}{5}} \cdot 27 + 3^{4})}{625}$

Step 5.2.4.14.17

Multiply $27$ by $20$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 540 x^{\frac{1}{5}} + 3^{4})}{625}$

Step 5.2.4.14.18

Raise $3$ to the power of $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}} + 1500 x^{\frac{3}{5}} + 1350 x^{\frac{2}{5}} + 540 x^{\frac{1}{5}} + 81)}{625}$

Step 5.2.4.15

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 3 (625 x^{\frac{4}{5}}) - 3 (1500 x^{\frac{3}{5}}) - 3 (1350 x^{\frac{2}{5}}) - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Step 5.2.4.16

Simplify.

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Step 5.2.4.16.1

Multiply $625$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 3 (1500 x^{\frac{3}{5}}) - 3 (1350 x^{\frac{2}{5}}) - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Step 5.2.4.16.2

Multiply $1500$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 3 (1350 x^{\frac{2}{5}}) - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Step 5.2.4.16.3

Multiply $1350$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 3 (540 x^{\frac{1}{5}}) - 3 \cdot 81}{625}$

Step 5.2.4.16.4

Multiply $540$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 3 \cdot 81}{625}$

Step 5.2.4.16.5

Multiply $- 3$ by $81$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Step 5.2.5

Simplify by adding terms.

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Step 5.2.5.1

Combine the opposite terms in $405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 486 + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 486 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243$ .

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Step 5.2.5.1.1

Add $- 486$ and $486$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} + 0 - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Step 5.2.5.1.2

Add $405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 4050 x^{\frac{2}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 4050 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Step 5.2.5.1.3

Subtract $4050 x^{\frac{2}{5}}$ from $4050 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} + 0 - 1620 x^{\frac{1}{5}} - 243}{625}$

Step 5.2.5.1.4

Add $405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}}$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} + 243 - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}} - 243}{625}$

Step 5.2.5.1.5

Subtract $243$ from $243$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}} + 0}{625}$

Step 5.2.5.1.6

Add $405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1620 x^{\frac{1}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}}$ and $0$ .

Step 5.2.5.2

Subtract $1620 x^{\frac{1}{5}}$ from $405 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 1215 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} + 2250 x^{\frac{3}{5}} + 2430 x^{\frac{1}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}}}{625}$

Step 5.2.5.3

Add $- 1215 x^{\frac{1}{5}}$ and $2430 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{1215 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} + 2250 x^{\frac{3}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}} - 1620 x^{\frac{1}{5}}}{625}$

Step 5.2.5.4

Subtract $1620 x^{\frac{1}{5}}$ from $1215 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} + 2250 x^{\frac{3}{5}} - 1875 x^{\frac{4}{5}} - 4500 x^{\frac{3}{5}}}{625}$

Step 5.2.5.5

Subtract $4500 x^{\frac{3}{5}}$ from $2250 x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Step 5.2.6

Simplify each term.

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Step 5.2.6.1

Move the negative in front of the fraction.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Step 5.2.6.2

Factor $- 15 x^{\frac{1}{5}}$ out of $- 405 x^{\frac{1}{5}} - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}$ .

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Step 5.2.6.2.1

Factor $- 15 x^{\frac{1}{5}}$ out of $- 405 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} \cdot 27 - 1350 x^{\frac{2}{5}} - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Step 5.2.6.2.2

Factor $- 15 x^{\frac{1}{5}}$ out of $- 1350 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} \cdot 27 - 15 x^{\frac{1}{5}} (90 x^{\frac{1}{5}}) - 1875 x^{\frac{4}{5}} - 2250 x^{\frac{3}{5}}}{625}$

Step 5.2.6.2.3

Factor $- 15 x^{\frac{1}{5}}$ out of $- 1875 x^{\frac{4}{5}}$ .

Step 5.2.6.2.4

Factor $- 15 x^{\frac{1}{5}}$ out of $- 2250 x^{\frac{3}{5}}$ .

Step 5.2.6.2.5

Factor $- 15 x^{\frac{1}{5}}$ out of $- 15 x^{\frac{1}{5}} (27) - 15 x^{\frac{1}{5}} (90 x^{\frac{1}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}}) - 15 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 15 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{625}$

Step 5.2.6.2.6

Factor $- 15 x^{\frac{1}{5}}$ out of $- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}}) - 15 x^{\frac{1}{5}} (125 x^{\frac{3}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}}) - 15 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{625}$

Step 5.2.6.2.7

Factor $- 15 x^{\frac{1}{5}}$ out of $- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}}) - 15 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})$ .

Step 5.2.6.3

Factor $5$ out of $- 15 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{5 (- 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}}))}{625}$

Step 5.2.6.4

Cancel the common factors.

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Step 5.2.6.4.1

Factor $5$ out of $625$ .

Step 5.2.6.4.2

Cancel the common factor.

Step 5.2.6.4.3

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} + \frac{- 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.6.5

Move the negative in front of the fraction.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = - \frac{243}{3125} + \frac{{(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.7

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + {(5 x^{\frac{1}{5}} + 3)}^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8

Simplify each term.

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Step 5.2.8.1

Simplify the numerator.

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Step 5.2.8.1.1

Use the Binomial Theorem.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + {(5 x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2

Simplify each term.

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Step 5.2.8.1.2.1

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 5^{5} {(x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 {(x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.3

Multiply the exponents in ${(x^{\frac{1}{5}})}^{5}$ .

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Step 5.2.8.1.2.3.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x^{\frac{1}{5} \cdot 5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.3.2

Cancel the common factor of $5$ .

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Step 5.2.8.1.2.3.2.1

Cancel the common factor.

Step 5.2.8.1.2.3.2.2

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.4

Simplify.

Step 5.2.8.1.2.5

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5 (5^{4} {(x^{\frac{1}{5}})}^{4}) \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.6

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

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Step 5.2.8.1.2.6.1

Move $5^{4}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5^{4} \cdot (5 {(x^{\frac{1}{5}})}^{4}) \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.6.2

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

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Step 5.2.8.1.2.6.2.1

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

Step 5.2.8.1.2.6.2.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5^{4 + 1} {(x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.6.3

Add $4$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 5^{5} {(x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.7

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 3125 {(x^{\frac{1}{5}})}^{4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.8

Multiply the exponents in ${(x^{\frac{1}{5}})}^{4}$ .

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Step 5.2.8.1.2.8.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 3125 x^{\frac{1}{5} \cdot 4} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.8.2

Combine $\frac{1}{5}$ and $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 3125 x^{\frac{4}{5}} \cdot 3 + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.9

Multiply $3$ by $3125$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 {(5 x^{\frac{1}{5}})}^{3} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.10

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (5^{3} {(x^{\frac{1}{5}})}^{3}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.11

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (125 {(x^{\frac{1}{5}})}^{3}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.12

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

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Step 5.2.8.1.2.12.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (125 x^{\frac{1}{5} \cdot 3}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.12.2

Combine $\frac{1}{5}$ and $3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 10 (125 x^{\frac{3}{5}}) \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.13

Multiply $125$ by $10$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} \cdot 3^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.14

Raise $3$ to the power of $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} \cdot 9 + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.15

Multiply $9$ by $1250$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 {(5 x^{\frac{1}{5}})}^{2} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.16

Apply the product rule to $5 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.17

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (25 {(x^{\frac{1}{5}})}^{2}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.18

Multiply the exponents in ${(x^{\frac{1}{5}})}^{2}$ .

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Step 5.2.8.1.2.18.1

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (25 x^{\frac{1}{5} \cdot 2}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.18.2

Combine $\frac{1}{5}$ and $2$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 10 (25 x^{\frac{2}{5}}) \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.19

Multiply $25$ by $10$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 250 x^{\frac{2}{5}} \cdot 3^{3} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.20

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 250 x^{\frac{2}{5}} \cdot 27 + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.21

Multiply $27$ by $250$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 5 (5 x^{\frac{1}{5}}) \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.22

Multiply $5$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} \cdot 3^{4} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.23

Raise $3$ to the power of $4$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} \cdot 81 + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.24

Multiply $81$ by $25$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}} + 3^{5}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.2.25

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{- 243 + 3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}} + 243}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.3

Add $- 243$ and $243$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}} + 0}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.4

Add $3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{3125 x + 9375 x^{\frac{4}{5}} + 11250 x^{\frac{3}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.5

Reorder terms.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{3125 x + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6

Rewrite $3125 x + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}$ in a factored form.

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Step 5.2.8.1.6.1

Factor $25 x^{\frac{1}{5}}$ out of $3125 x + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}$ .

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Step 5.2.8.1.6.1.1

Factor $25 x^{\frac{1}{5}}$ out of $3125 x$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 11250 x^{\frac{3}{5}} + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.2

Factor $25 x^{\frac{1}{5}}$ out of $11250 x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 9375 x^{\frac{4}{5}} + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.3

Factor $25 x^{\frac{1}{5}}$ out of $9375 x^{\frac{4}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 6750 x^{\frac{2}{5}} + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.4

Factor $25 x^{\frac{1}{5}}$ out of $6750 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 2025 x^{\frac{1}{5}}}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.5

Factor $25 x^{\frac{1}{5}}$ out of $2025 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.6

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.7

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (375 x^{\frac{3}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.8

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}}) + 25 x^{\frac{1}{5}} (270 x^{\frac{1}{5}})$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.1.9

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (81)$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.1.6.2

Reorder terms.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81)}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.2

Factor $25$ out of $25 x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81)$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 (x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81))}{3125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.3

Cancel the common factors.

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Step 5.2.8.3.1

Factor $25$ out of $3125$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{25 (x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81))}{25 \cdot 125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.8.3.2

Cancel the common factor.

Step 5.2.8.3.3

Rewrite the expression.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81)}{125} - \frac{3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.9

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{x^{\frac{1}{5}} (450 x^{\frac{2}{5}} + 375 x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} + 125 x^{\frac{4}{5}} + 81) - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10

Simplify each term.

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Step 5.2.10.1

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{x^{\frac{1}{5}} (450 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.2

Simplify.

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Step 5.2.10.2.1

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + x^{\frac{1}{5}} (375 x^{\frac{3}{5}}) + x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.2.2

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} (270 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.2.3

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.2.4

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + x^{\frac{1}{5}} \cdot 81 - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.2.5

Move $81$ to the left of $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{1}{5}} x^{\frac{2}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3

Simplify each term.

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Step 5.2.10.3.1

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{2}{5}}$ by adding the exponents.

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Step 5.2.10.3.1.1

Move $x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 (x^{\frac{2}{5}} x^{\frac{1}{5}}) + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.1.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{2}{5} + \frac{1}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.1.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{2 + 1}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.1.4

Add $2$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{1}{5}} x^{\frac{3}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.2

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{3}{5}}$ by adding the exponents.

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Step 5.2.10.3.2.1

Move $x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 (x^{\frac{3}{5}} x^{\frac{1}{5}}) + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.2.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{3}{5} + \frac{1}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.2.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{3 + 1}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.2.4

Add $3$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{1}{5}} x^{\frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.3

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 5.2.10.3.3.1

Move $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 (x^{\frac{1}{5}} x^{\frac{1}{5}}) + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.3.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{1}{5} + \frac{1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.3.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{1 + 1}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.3.4

Add $1$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.4

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{4}{5}}$ by adding the exponents.

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Step 5.2.10.3.4.1

Move $x^{\frac{4}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 (x^{\frac{4}{5}} x^{\frac{1}{5}}) + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.4.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{4}{5} + \frac{1}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.4.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{4 + 1}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.4.4

Add $4$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x^{\frac{5}{5}} + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.3.4.5

Divide $5$ by $5$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 \cdot x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (27 + 90 x^{\frac{1}{5}} + 125 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.4

Apply the distributive property.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} \cdot 27 - 3 x^{\frac{1}{5}} (90 x^{\frac{1}{5}}) - 3 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.5

Simplify.

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Step 5.2.10.5.1

Multiply $27$ by $- 3$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 x^{\frac{1}{5}} (90 x^{\frac{1}{5}}) - 3 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.5.2

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5}} x^{\frac{1}{5}}) - 3 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.5.3

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5}} x^{\frac{1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 x^{\frac{1}{5}} (150 x^{\frac{2}{5}})}{125}$

Step 5.2.10.5.4

Rewrite using the commutative property of multiplication.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5}} x^{\frac{1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6

Simplify each term.

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Step 5.2.10.6.1

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{1}{5}}$ by adding the exponents.

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Step 5.2.10.6.1.1

Move $x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 (x^{\frac{1}{5}} x^{\frac{1}{5}})) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.1.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1}{5} + \frac{1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.1.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{1 + 1}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.1.4

Add $1$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 3 \cdot (90 x^{\frac{2}{5}}) - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.2

Multiply $- 3$ by $90$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{1}{5}} x^{\frac{3}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.3

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{3}{5}}$ by adding the exponents.

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Step 5.2.10.6.3.1

Move $x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 (x^{\frac{3}{5}} x^{\frac{1}{5}})) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.3.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{3}{5} + \frac{1}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.3.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{3 + 1}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.3.4

Add $3$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 3 \cdot (125 x^{\frac{4}{5}}) - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.4

Multiply $- 3$ by $125$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{1}{5}} x^{\frac{2}{5}})}{125}$

Step 5.2.10.6.5

Multiply $x^{\frac{1}{5}}$ by $x^{\frac{2}{5}}$ by adding the exponents.

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Step 5.2.10.6.5.1

Move $x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 (x^{\frac{2}{5}} x^{\frac{1}{5}}))}{125}$

Step 5.2.10.6.5.2

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

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{2}{5} + \frac{1}{5}})}{125}$

Step 5.2.10.6.5.3

Combine the numerators over the common denominator.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{2 + 1}{5}})}{125}$

Step 5.2.10.6.5.4

Add $2$ and $1$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 3 \cdot (150 x^{\frac{3}{5}})}{125}$

Step 5.2.10.6.6

Multiply $- 3$ by $150$ .

Step 5.2.11

Simplify terms.

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Step 5.2.11.1

Combine the opposite terms in $450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 81 x^{\frac{1}{5}} - 81 x^{\frac{1}{5}} - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}$ .

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Step 5.2.11.1.1

Subtract $81 x^{\frac{1}{5}}$ from $81 x^{\frac{1}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x + 0 - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Step 5.2.11.1.2

Add $450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 270 x^{\frac{2}{5}} + 125 x - 270 x^{\frac{2}{5}} - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Step 5.2.11.1.3

Subtract $270 x^{\frac{2}{5}}$ from $270 x^{\frac{2}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 125 x + 0 - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Step 5.2.11.1.4

Add $450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 375 x^{\frac{4}{5}} + 125 x - 375 x^{\frac{4}{5}} - 450 x^{\frac{3}{5}}}{125}$

Step 5.2.11.1.5

Subtract $375 x^{\frac{4}{5}}$ from $375 x^{\frac{4}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 125 x + 0 - 450 x^{\frac{3}{5}}}{125}$

Step 5.2.11.1.6

Add $450 x^{\frac{3}{5}} + 125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{450 x^{\frac{3}{5}} + 125 x - 450 x^{\frac{3}{5}}}{125}$

Step 5.2.11.1.7

Subtract $450 x^{\frac{3}{5}}$ from $450 x^{\frac{3}{5}}$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{125 x + 0}{125}$

Step 5.2.11.1.8

Add $125 x$ and $0$ .

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{125 x}{125}$

Step 5.2.11.2

Cancel the common factor of $125$ .

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Step 5.2.11.2.1

Cancel the common factor.

$f^{- 1} (5 x^{\frac{1}{5}} + 3) = \frac{125 x}{125}$

Step 5.2.11.2.2

Divide $x$ by $1$ .

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

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 $f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125})}^{\frac{1}{5}} + 3$

Step 5.3.3

Move $- \frac{243}{3125}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(\frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Step 5.3.4

Move $\frac{81 x}{625}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(- \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Step 5.3.5

Move $- \frac{54 x^{2}}{625}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(\frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} - \frac{54 x^{2}}{625} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Step 5.3.6

Move $\frac{18 x^{3}}{625}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(- \frac{3 x^{4}}{625} + \frac{x^{5}}{3125} + \frac{18 x^{3}}{625} - \frac{54 x^{2}}{625} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Step 5.3.7

Reorder $- \frac{3 x^{4}}{625}$ and $\frac{x^{5}}{3125}$ .

$f (- \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}) = 5 {(\frac{x^{5}}{3125} - \frac{3 x^{4}}{625} + \frac{18 x^{3}}{625} - \frac{54 x^{2}}{625} + \frac{81 x}{625} - \frac{243}{3125})}^{\frac{1}{5}} + 3$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$ is the inverse of $f (x) = 5 x^{\frac{1}{5}} + 3$ .

$f^{- 1} (x) = - \frac{243}{3125} + \frac{81 x}{625} - \frac{54 x^{2}}{625} + \frac{18 x^{3}}{625} - \frac{3 x^{4}}{625} + \frac{x^{5}}{3125}$