Algebra Examples

$f (x) = \sqrt[5]{\frac{x^{3}}{9}}$

Step 1

Write $f (x) = \sqrt[5]{\frac{x^{3}}{9}}$ as an equation.

$y = \sqrt[5]{\frac{x^{3}}{9}}$

Step 2

Interchange the variables.

$x = \sqrt[5]{\frac{y^{3}}{9}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[5]{\frac{y^{3}}{9}} = x$ .

$\sqrt[5]{\frac{y^{3}}{9}} = x$

Step 3.2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{\frac{y^{3}}{9}}}^{5} = x^{5}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{\frac{y^{3}}{9}}$ as ${(\frac{y^{3}}{9})}^{\frac{1}{5}}$ .

${({(\frac{y^{3}}{9})}^{\frac{1}{5}})}^{5} = x^{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({(\frac{y^{3}}{9})}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({(\frac{y^{3}}{9})}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.1.1.1

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

${(\frac{y^{3}}{9})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${(\frac{y^{3}}{9})}^{\frac{1}{5} \cdot 5} = x^{5}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${(\frac{y^{3}}{9})}^{1} = x^{5}$

Step 3.3.2.1.2

Simplify.

$\frac{y^{3}}{9} = x^{5}$

Step 3.4

Solve for $y$ .

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Step 3.4.1

Multiply both sides of the equation by $9$ .

$9 \frac{y^{3}}{9} = 9 x^{5}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $9$ .

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Step 3.4.2.1.1

Cancel the common factor.

$9 \frac{y^{3}}{9} = 9 x^{5}$

Step 3.4.2.1.2

Rewrite the expression.

$y^{3} = 9 x^{5}$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{9 x^{5}}$

Step 3.4.4

Simplify $\sqrt[3]{9 x^{5}}$ .

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Step 3.4.4.1

Rewrite $9 x^{5}$ as $x^{3} \cdot (9 x^{2})$ .

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Step 3.4.4.1.1

Factor out $x^{3}$ .

$y = \sqrt[3]{9 (x^{3} x^{2})}$

Step 3.4.4.1.2

Reorder $9$ and $x^{3}$ .

$y = \sqrt[3]{x^{3} \cdot 9 x^{2}}$

Step 3.4.4.1.3

Add parentheses.

$y = \sqrt[3]{x^{3} \cdot (9 x^{2})}$

Step 3.4.4.2

Pull terms out from under the radical.

$y = x \sqrt[3]{9 x^{2}}$

Step 4

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

$f^{- 1} (x) = x \sqrt[3]{9 x^{2}}$

Step 5

Verify if $f^{- 1} (x) = x \sqrt[3]{9 x^{2}}$ is the inverse of $f (x) = \sqrt[5]{\frac{x^{3}}{9}}$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = (\sqrt[5]{\frac{x^{3}}{9}}) \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.3

Rewrite $\sqrt[5]{\frac{x^{3}}{9}}$ as $\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.4

Multiply $\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}}$ by $\frac{{\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{4}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}} \cdot \frac{{\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{4}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5

Combine and simplify the denominator.

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Step 5.2.5.1

Multiply $\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}}$ by $\frac{{\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{4}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{\sqrt[5]{9} {\sqrt[5]{9}}^{4}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.2

Raise $\sqrt[5]{9}$ to the power of $1$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{\sqrt[5]{9} {\sqrt[5]{9}}^{4}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.3

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{1 + 4}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.4

Add $1$ and $4$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{5}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.5

Rewrite ${\sqrt[5]{9}}^{5}$ as $9$ .

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Step 5.2.5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{9}$ as $9^{\frac{1}{5}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{{(9^{\frac{1}{5}})}^{5}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.5.2

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9^{\frac{1}{5} \cdot 5}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.5.3

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9^{\frac{5}{5}}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.5.4

Cancel the common factor of $5$ .

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Step 5.2.5.5.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9^{\frac{5}{5}}} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.5.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.5.5.5

Evaluate the exponent.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.6

Simplify the numerator.

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Step 5.2.6.1

Rewrite ${\sqrt[5]{9}}^{4}$ as $\sqrt[5]{9^{4}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} \sqrt[5]{9^{4}}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.6.2

Raise $9$ to the power of $4$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} \sqrt[5]{6561}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.6.3

Rewrite $6561$ as $3^{5} \cdot 27$ .

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Step 5.2.6.3.1

Factor $243$ out of $6561$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} \sqrt[5]{243 (27)}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.6.3.2

Rewrite $243$ as $3^{5}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} \sqrt[5]{3^{5} \cdot 27}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.6.4

Pull terms out from under the radical.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3}} \cdot (3 \sqrt[5]{27})}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.6.5

Combine using the product rule for radicals.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{3 \sqrt[5]{x^{3} \cdot 27}}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.7

Cancel the common factor of $3$ and $9$ .

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Step 5.2.7.1

Factor $3$ out of $3 \sqrt[5]{x^{3} \cdot 27}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{3 (\sqrt[5]{x^{3} \cdot 27})}{9} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.7.2

Cancel the common factors.

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Step 5.2.7.2.1

Factor $3$ out of $9$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{3 \sqrt[5]{x^{3} \cdot 27}}{3 \cdot 3} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.7.2.2

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{3 \sqrt[5]{x^{3} \cdot 27}}{3 \cdot 3} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.7.2.3

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\sqrt[5]{\frac{x^{3}}{9}})}^{2}}$

Step 5.2.8

Rewrite $\sqrt[5]{\frac{x^{3}}{9}}$ as $\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}})}^{2}}$

Step 5.2.9

Multiply $\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}}$ by $\frac{{\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{4}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}} \cdot \frac{{\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{4}})}^{2}}$

Step 5.2.10

Combine and simplify the denominator.

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Step 5.2.10.1

Multiply $\frac{\sqrt[5]{x^{3}}}{\sqrt[5]{9}}$ by $\frac{{\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{4}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{\sqrt[5]{9} {\sqrt[5]{9}}^{4}})}^{2}}$

Step 5.2.10.2

Raise $\sqrt[5]{9}$ to the power of $1$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{\sqrt[5]{9} {\sqrt[5]{9}}^{4}})}^{2}}$

Step 5.2.10.3

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{1 + 4}})}^{2}}$

Step 5.2.10.4

Add $1$ and $4$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{{\sqrt[5]{9}}^{5}})}^{2}}$

Step 5.2.10.5

Rewrite ${\sqrt[5]{9}}^{5}$ as $9$ .

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Step 5.2.10.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{9}$ as $9^{\frac{1}{5}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{{(9^{\frac{1}{5}})}^{5}})}^{2}}$

Step 5.2.10.5.2

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9^{\frac{1}{5} \cdot 5}})}^{2}}$

Step 5.2.10.5.3

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9^{\frac{5}{5}}})}^{2}}$

Step 5.2.10.5.4

Cancel the common factor of $5$ .

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Step 5.2.10.5.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9^{\frac{5}{5}}})}^{2}}$

Step 5.2.10.5.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9})}^{2}}$

Step 5.2.10.5.5

Evaluate the exponent.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} {\sqrt[5]{9}}^{4}}{9})}^{2}}$

Step 5.2.11

Simplify the numerator.

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Step 5.2.11.1

Rewrite ${\sqrt[5]{9}}^{4}$ as $\sqrt[5]{9^{4}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} \sqrt[5]{9^{4}}}{9})}^{2}}$

Step 5.2.11.2

Raise $9$ to the power of $4$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} \sqrt[5]{6561}}{9})}^{2}}$

Step 5.2.11.3

Rewrite $6561$ as $3^{5} \cdot 27$ .

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Step 5.2.11.3.1

Factor $243$ out of $6561$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} \sqrt[5]{243 (27)}}{9})}^{2}}$

Step 5.2.11.3.2

Rewrite $243$ as $3^{5}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} \sqrt[5]{3^{5} \cdot 27}}{9})}^{2}}$

Step 5.2.11.4

Pull terms out from under the radical.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3}} \cdot (3 \sqrt[5]{27})}{9})}^{2}}$

Step 5.2.11.5

Combine using the product rule for radicals.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{3 \sqrt[5]{x^{3} \cdot 27}}{9})}^{2}}$

Step 5.2.12

Reduce the expression by cancelling the common factors.

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Step 5.2.12.1

Cancel the common factor of $3$ and $9$ .

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Step 5.2.12.1.1

Factor $3$ out of $3 \sqrt[5]{x^{3} \cdot 27}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{3 (\sqrt[5]{x^{3} \cdot 27})}{9})}^{2}}$

Step 5.2.12.1.2

Cancel the common factors.

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Step 5.2.12.1.2.1

Factor $3$ out of $9$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{3 \sqrt[5]{x^{3} \cdot 27}}{3 \cdot 3})}^{2}}$

Step 5.2.12.1.2.2

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{3 \sqrt[5]{x^{3} \cdot 27}}{3 \cdot 3})}^{2}}$

Step 5.2.12.1.2.3

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 {(\frac{\sqrt[5]{x^{3} \cdot 27}}{3})}^{2}}$

Step 5.2.12.2

Apply the product rule to $\frac{\sqrt[5]{x^{3} \cdot 27}}{3}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{{\sqrt[5]{x^{3} \cdot 27}}^{2}}{3^{2}})}$

Step 5.2.13

Simplify the numerator.

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Step 5.2.13.1

Rewrite ${\sqrt[5]{x^{3} \cdot 27}}^{2}$ as $\sqrt[5]{{(x^{3} \cdot 27)}^{2}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{{(x^{3} \cdot 27)}^{2}}}{3^{2}})}$

Step 5.2.13.2

Apply the product rule to $x^{3} \cdot 27$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{{(x^{3})}^{2} \cdot 27^{2}}}{3^{2}})}$

Step 5.2.13.3

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 5.2.13.3.1

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{3 \cdot 2} \cdot 27^{2}}}{3^{2}})}$

Step 5.2.13.3.2

Multiply $3$ by $2$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{6} \cdot 27^{2}}}{3^{2}})}$

Step 5.2.13.4

Raise $27$ to the power of $2$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{6} \cdot 729}}{3^{2}})}$

Step 5.2.13.5

Rewrite $x^{6} \cdot 729$ as ${(x \cdot 3)}^{5} (x \cdot 3)$ .

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Step 5.2.13.5.1

Factor out $x^{5}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{5} x \cdot 729}}{3^{2}})}$

Step 5.2.13.5.2

Factor $243$ out of $729$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{5} x \cdot (243 (3))}}{3^{2}})}$

Step 5.2.13.5.3

Rewrite $243$ as $3^{5}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{5} x \cdot (3^{5} \cdot 3)}}{3^{2}})}$

Step 5.2.13.5.4

Move $x$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{x^{5} \cdot (3^{5} x) \cdot 3}}{3^{2}})}$

Step 5.2.13.5.5

Rewrite $x^{5} \cdot 3^{5}$ as ${(x \cdot 3)}^{5}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{{(x \cdot 3)}^{5} x \cdot 3}}{3^{2}})}$

Step 5.2.13.5.6

Add parentheses.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{\sqrt[5]{{(x \cdot 3)}^{5} (x \cdot 3)}}{3^{2}})}$

Step 5.2.13.6

Pull terms out from under the radical.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{x \cdot (3 \sqrt[5]{x \cdot 3})}{3^{2}})}$

Step 5.2.13.7

Move $3$ to the left of $x$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{3 x \sqrt[5]{x \cdot 3}}{3^{2}})}$

Step 5.2.14

Simplify terms.

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Step 5.2.14.1

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{3 x \sqrt[5]{x \cdot 3}}{9})}$

Step 5.2.14.2

Cancel the common factor of $3$ and $9$ .

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Step 5.2.14.2.1

Factor $3$ out of $3 x \sqrt[5]{x \cdot 3}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{3 (x \sqrt[5]{x \cdot 3})}{9})}$

Step 5.2.14.2.2

Cancel the common factors.

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Step 5.2.14.2.2.1

Factor $3$ out of $9$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{3 (x \sqrt[5]{x \cdot 3})}{3 \cdot 3})}$

Step 5.2.14.2.2.2

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{3 (x \sqrt[5]{x \cdot 3})}{3 \cdot 3})}$

Step 5.2.14.2.2.3

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{x \sqrt[5]{x \cdot 3}}{3})}$

Step 5.2.14.3

Move $3$ to the left of $x$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{9 (\frac{x \sqrt[5]{3 \cdot x}}{3})}$

Step 5.2.14.4

Combine $9$ and $\frac{x \sqrt[5]{3 x}}{3}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{\frac{9 (x \sqrt[5]{3 x})}{3}}$

Step 5.2.15

Reduce the expression by cancelling the common factors.

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Step 5.2.15.1

Reduce the expression $\frac{9 x \sqrt[5]{3 x}}{3}$ by cancelling the common factors.

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Step 5.2.15.1.1

Factor $3$ out of $9 x \sqrt[5]{3 x}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{\frac{3 (3 x \sqrt[5]{3 x})}{3}}$

Step 5.2.15.1.2

Factor $3$ out of $3$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{\frac{3 (3 x \sqrt[5]{3 x})}{3 (1)}}$

Step 5.2.15.1.3

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{\frac{3 (3 x \sqrt[5]{3 x})}{3 \cdot 1}}$

Step 5.2.15.1.4

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{\frac{3 x \sqrt[5]{3 x}}{1}}$

Step 5.2.15.2

Divide $3 x \sqrt[5]{3 x}$ by $1$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27}}{3} \cdot \sqrt[3]{3 x \sqrt[5]{3 x}}$

Step 5.2.16

Multiply $\frac{\sqrt[5]{x^{3} \cdot 27}}{3} \sqrt[3]{3 x \sqrt[5]{3 x}}$ .

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Step 5.2.16.1

Combine $\frac{\sqrt[5]{x^{3} \cdot 27}}{3}$ and $\sqrt[3]{3 x \sqrt[5]{3 x}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[5]{x^{3} \cdot 27} \sqrt[3]{3 x \sqrt[5]{3 x}}}{3}$

Step 5.2.16.2

Rewrite the expression using the least common index of $15$ .

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Step 5.2.16.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{3} \cdot 27}$ as ${(x^{3} \cdot 27)}^{\frac{1}{5}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{{(x^{3} \cdot 27)}^{\frac{1}{5}} \sqrt[3]{3 x \sqrt[5]{3 x}}}{3}$

Step 5.2.16.2.2

Rewrite ${(x^{3} \cdot 27)}^{\frac{1}{5}}$ as ${(x^{3} \cdot 27)}^{\frac{3}{15}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{{(x^{3} \cdot 27)}^{\frac{3}{15}} \sqrt[3]{3 x \sqrt[5]{3 x}}}{3}$

Step 5.2.16.2.3

Rewrite ${(x^{3} \cdot 27)}^{\frac{3}{15}}$ as $\sqrt[15]{{(x^{3} \cdot 27)}^{3}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3} \cdot 27)}^{3}} \sqrt[3]{3 x \sqrt[5]{3 x}}}{3}$

Step 5.2.16.2.4

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3} \cdot 27)}^{3}} {(3 x \sqrt[5]{3 x})}^{\frac{1}{3}}}{3}$

Step 5.2.16.2.5

Rewrite ${(3 x \sqrt[5]{3 x})}^{\frac{1}{3}}$ as ${(3 x \sqrt[5]{3 x})}^{\frac{5}{15}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3} \cdot 27)}^{3}} {(3 x \sqrt[5]{3 x})}^{\frac{5}{15}}}{3}$

Step 5.2.16.2.6

Rewrite ${(3 x \sqrt[5]{3 x})}^{\frac{5}{15}}$ as $\sqrt[15]{{(3 x \sqrt[5]{3 x})}^{5}}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3} \cdot 27)}^{3}} \sqrt[15]{{(3 x \sqrt[5]{3 x})}^{5}}}{3}$

Step 5.2.16.3

Combine using the product rule for radicals.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3} \cdot 27)}^{3} {(3 x \sqrt[5]{3 x})}^{5}}}{3}$

Step 5.2.16.4

Apply the product rule to $x^{3} \cdot 27$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3})}^{3} \cdot (27^{3} {(3 x \sqrt[5]{3 x})}^{5})}}{3}$

Step 5.2.16.5

Apply the product rule to $3 x \sqrt[5]{3 x}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(x^{3})}^{3} \cdot (27^{3} {(3 x)}^{5} {\sqrt[5]{3 x}}^{5})}}{3}$

Step 5.2.16.6

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 5.2.16.6.1

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{3 \cdot 3} \cdot (27^{3} {(3 x)}^{5} {\sqrt[5]{3 x}}^{5})}}{3}$

Step 5.2.16.6.2

Multiply $3$ by $3$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (27^{3} {(3 x)}^{5} {\sqrt[5]{3 x}}^{5})}}{3}$

Step 5.2.16.7

Raise $27$ to the power of $3$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 {(3 x)}^{5} {\sqrt[5]{3 x}}^{5})}}{3}$

Step 5.2.16.8

Apply the product rule to $3 x$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (3^{5} x^{5}) {\sqrt[5]{3 x}}^{5})}}{3}$

Step 5.2.16.9

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) {\sqrt[5]{3 x}}^{5})}}{3}$

Step 5.2.16.10

Rewrite ${\sqrt[5]{3 x}}^{5}$ as $3 x$ .

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Step 5.2.16.10.1

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) {({(3 x)}^{\frac{1}{5}})}^{5})}}{3}$

Step 5.2.16.10.2

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) {(3 x)}^{\frac{1}{5} \cdot 5})}}{3}$

Step 5.2.16.10.3

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) {(3 x)}^{\frac{5}{5}})}}{3}$

Step 5.2.16.10.4

Cancel the common factor of $5$ .

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Step 5.2.16.10.4.1

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) {(3 x)}^{\frac{5}{5}})}}{3}$

Step 5.2.16.10.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) (3 x))}}{3}$

Step 5.2.16.10.5

Simplify.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (19683 (243 x^{5}) (3 x))}}{3}$

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot 19683 \cdot (243 x^{5}) \cdot (3 x)}}{3}$

Step 5.2.16.11

Combine exponents.

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Step 5.2.16.11.1

Multiply $243$ by $19683$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{9} \cdot (4782969 x^{5}) \cdot (3 x)}}{3}$

Step 5.2.16.11.2

Multiply $x^{9}$ by $x^{5}$ by adding the exponents.

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Step 5.2.16.11.2.1

Move $x^{5}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{5} x^{9} \cdot 4782969 \cdot (3 x)}}{3}$

Step 5.2.16.11.2.2

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{5 + 9} \cdot 4782969 \cdot (3 x)}}{3}$

Step 5.2.16.11.2.3

Add $5$ and $9$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{14} \cdot 4782969 \cdot (3 x)}}{3}$

Step 5.2.16.11.3

Multiply $3$ by $4782969$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{x^{14} \cdot (14348907 x)}}{3}$

Step 5.2.16.11.4

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{14348907 (x^{14} x)}}{3}$

Step 5.2.16.11.5

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{14348907 x^{14 + 1}}}{3}$

Step 5.2.16.11.6

Add $14$ and $1$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{14348907 x^{15}}}{3}$

Step 5.2.17

Simplify the numerator.

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Step 5.2.17.1

Rewrite $14348907 x^{15}$ as ${(3 x)}^{15}$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{\sqrt[15]{{(3 x)}^{15}}}{3}$

Step 5.2.17.2

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

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{3 x}{3}$

Step 5.2.18

Cancel the common factor of $3$ .

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Step 5.2.18.1

Cancel the common factor.

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = \frac{3 x}{3}$

Step 5.2.18.2

Divide $x$ by $1$ .

$f^{- 1} (\sqrt[5]{\frac{x^{3}}{9}}) = x$

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 $f (x \sqrt[3]{9 x^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{{(x \sqrt[3]{9 x^{2}})}^{3}}{9}}$

Step 5.3.3

Apply the product rule to $x \sqrt[3]{9 x^{2}}$ .

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} {\sqrt[3]{9 x^{2}}}^{3}}{9}}$

Step 5.3.4

Simplify the numerator.

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Step 5.3.4.1

Rewrite ${\sqrt[3]{9 x^{2}}}^{3}$ as $9 x^{2}$ .

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Step 5.3.4.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x^{2}}$ as ${(9 x^{2})}^{\frac{1}{3}}$ .

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} {({(9 x^{2})}^{\frac{1}{3}})}^{3}}{9}}$

Step 5.3.4.1.2

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

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} {(9 x^{2})}^{\frac{1}{3} \cdot 3}}{9}}$

Step 5.3.4.1.3

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

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} {(9 x^{2})}^{\frac{3}{3}}}{9}}$

Step 5.3.4.1.4

Cancel the common factor of $3$ .

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Step 5.3.4.1.4.1

Cancel the common factor.

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} {(9 x^{2})}^{\frac{3}{3}}}{9}}$

Step 5.3.4.1.4.2

Rewrite the expression.

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} (9 x^{2})}{9}}$

Step 5.3.4.1.5

Simplify.

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} (9 x^{2})}{9}}$

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{3} \cdot (9 x^{2})}{9}}$

Step 5.3.4.2

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

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Step 5.3.4.2.1

Move $x^{2}$ .

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{2} x^{3} \cdot 9}{9}}$

Step 5.3.4.2.2

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

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{2 + 3} \cdot 9}{9}}$

Step 5.3.4.2.3

Add $2$ and $3$ .

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{5} \cdot 9}{9}}$

Step 5.3.5

Cancel the common factor of $9$ .

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Step 5.3.5.1

Cancel the common factor.

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{\frac{x^{5} \cdot 9}{9}}$

Step 5.3.5.2

Divide $x^{5}$ by $1$ .

$f (x \sqrt[3]{9 x^{2}}) = \sqrt[5]{x^{5}}$

Step 5.3.6

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

$f (x \sqrt[3]{9 x^{2}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x \sqrt[3]{9 x^{2}}$ is the inverse of $f (x) = \sqrt[5]{\frac{x^{3}}{9}}$ .

$f^{- 1} (x) = x \sqrt[3]{9 x^{2}}$