Find the Inverse f(x)=9( fifth root of x+2)

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

Write $f (x) = 9 (\sqrt[5]{x} + 2)$ as an equation.

$y = 9 (\sqrt[5]{x} + 2)$

Interchange the variables.

$x = 9 (\sqrt[5]{y} + 2)$

Solve for $y$ .

Rewrite the equation as $9 (\sqrt[5]{y} + 2) = x$ .

$9 (\sqrt[5]{y} + 2) = x$

Divide each term in $9 (\sqrt[5]{y} + 2) = x$ by $9$ and simplify.

Divide each term in $9 (\sqrt[5]{y} + 2) = x$ by $9$ .

$\frac{9 (\sqrt[5]{y} + 2)}{9} = \frac{x}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 (\sqrt[5]{y} + 2)}{9} = \frac{x}{9}$

Divide $\sqrt[5]{y} + 2$ by $1$ .

$\sqrt[5]{y} + 2 = \frac{x}{9}$

Subtract $2$ from both sides of the equation.

$\sqrt[5]{y} = \frac{x}{9} - 2$

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

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

Simplify each side of the equation.

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

${(y^{\frac{1}{5}})}^{5} = {(\frac{x}{9} - 2)}^{5}$

Simplify the left side.

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

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

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

$y^{\frac{1}{5} \cdot 5} = {(\frac{x}{9} - 2)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = {(\frac{x}{9} - 2)}^{5}$

Rewrite the expression.

$y^{1} = {(\frac{x}{9} - 2)}^{5}$

Simplify.

$y = {(\frac{x}{9} - 2)}^{5}$

Simplify the right side.

Simplify ${(\frac{x}{9} - 2)}^{5}$ .

Use the Binomial Theorem.

$y = {(\frac{x}{9})}^{5} + 5 {(\frac{x}{9})}^{4} \cdot - 2 + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Simplify each term.

Apply the product rule to $\frac{x}{9}$ .

$y = \frac{x^{5}}{9^{5}} + 5 {(\frac{x}{9})}^{4} \cdot - 2 + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Raise $9$ to the power of $5$ .

$y = \frac{x^{5}}{59049} + 5 {(\frac{x}{9})}^{4} \cdot - 2 + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Apply the product rule to $\frac{x}{9}$ .

$y = \frac{x^{5}}{59049} + 5 \frac{x^{4}}{9^{4}} \cdot - 2 + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

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

$y = \frac{x^{5}}{59049} + 5 \frac{x^{4}}{6561} \cdot - 2 + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Combine $5$ and $\frac{x^{4}}{6561}$ .

$y = \frac{x^{5}}{59049} + \frac{5 x^{4}}{6561} \cdot - 2 + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $\frac{5 x^{4}}{6561} \cdot - 2$ .

Combine $\frac{5 x^{4}}{6561}$ and $- 2$ .

$y = \frac{x^{5}}{59049} + \frac{5 x^{4} \cdot - 2}{6561} + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $- 2$ by $5$ .

$y = \frac{x^{5}}{59049} + \frac{- 10 x^{4}}{6561} + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Move the negative in front of the fraction.

$y = \frac{x^{5}}{59049} - \frac{(10) x^{4}}{6561} + 10 {(\frac{x}{9})}^{3} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Apply the product rule to $\frac{x}{9}$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + 10 \frac{x^{3}}{9^{3}} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Raise $9$ to the power of $3$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + 10 \frac{x^{3}}{729} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Combine $10$ and $\frac{x^{3}}{729}$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{10 x^{3}}{729} {(- 2)}^{2} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

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

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{10 x^{3}}{729} \cdot 4 + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $\frac{10 x^{3}}{729} \cdot 4$ .

Combine $\frac{10 x^{3}}{729}$ and $4$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{10 x^{3} \cdot 4}{729} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $4$ by $10$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + 10 {(\frac{x}{9})}^{2} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Apply the product rule to $\frac{x}{9}$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + 10 \frac{x^{2}}{9^{2}} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Raise $9$ to the power of $2$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + 10 \frac{x^{2}}{81} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Combine $10$ and $\frac{x^{2}}{81}$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + \frac{10 x^{2}}{81} {(- 2)}^{3} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

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

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + \frac{10 x^{2}}{81} \cdot - 8 + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $\frac{10 x^{2}}{81} \cdot - 8$ .

Combine $\frac{10 x^{2}}{81}$ and $- 8$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + \frac{10 x^{2} \cdot - 8}{81} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Multiply $- 8$ by $10$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} + \frac{- 80 x^{2}}{81} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Move the negative in front of the fraction.

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{(80) x^{2}}{81} + 5 \frac{x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Combine $5$ and $\frac{x}{9}$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{5 x}{9} {(- 2)}^{4} + {(- 2)}^{5}$

Raise $- 2$ to the power of $4$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{5 x}{9} \cdot 16 + {(- 2)}^{5}$

Multiply $\frac{5 x}{9} \cdot 16$ .

Combine $\frac{5 x}{9}$ and $16$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{5 x \cdot 16}{9} + {(- 2)}^{5}$

Multiply $16$ by $5$ .

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} + {(- 2)}^{5}$

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

$y = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32$

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

$f^{- 1} (x) = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32$

Verify if $f^{- 1} (x) = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32$ is the inverse of $f (x) = 9 (\sqrt[5]{x} + 2)$ .

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

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

Set up the composite result function.

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

Evaluate $f^{- 1} (9 (\sqrt[5]{x} + 2))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = \frac{{(9 (\sqrt[5]{x} + 2))}^{5}}{59049} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify each term.

Simplify the numerator.

Apply the product rule to $9 (\sqrt[5]{x} + 2)$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = \frac{9^{5} {(\sqrt[5]{x} + 2)}^{5}}{59049} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Raise $9$ to the power of $5$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = \frac{59049 {(\sqrt[5]{x} + 2)}^{5}}{59049} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor of $59049$ .

Cancel the common factor.

Divide ${(\sqrt[5]{x} + 2)}^{5}$ by $1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = {(\sqrt[5]{x} + 2)}^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Use the Binomial Theorem.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = {\sqrt[5]{x}}^{5} + 5 {\sqrt[5]{x}}^{4} \cdot 2 + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify each term.

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

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = {(x^{\frac{1}{5}})}^{5} + 5 {\sqrt[5]{x}}^{4} \cdot 2 + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x^{\frac{1}{5} \cdot 5} + 5 {\sqrt[5]{x}}^{4} \cdot 2 + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x^{\frac{5}{5}} + 5 {\sqrt[5]{x}}^{4} \cdot 2 + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor of $5$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 5 {\sqrt[5]{x}}^{4} \cdot 2 + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify.

Rewrite ${\sqrt[5]{x}}^{4}$ as ${(x^{4})}^{\frac{1}{5}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 5 \sqrt[5]{x^{4}} \cdot 2 + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $2$ by $5$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 10 {\sqrt[5]{x}}^{3} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 10 \sqrt[5]{x^{3}} \cdot 2^{2} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 10 \sqrt[5]{x^{3}} \cdot 4 + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $4$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 10 {\sqrt[5]{x}}^{2} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite ${\sqrt[5]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{5}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 10 \sqrt[5]{x^{2}} \cdot 2^{3} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 10 \sqrt[5]{x^{2}} \cdot 8 + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $8$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 5 \sqrt[5]{x} \cdot 2^{4} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Raise $2$ to the power of $4$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 5 \sqrt[5]{x} \cdot 16 + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $16$ by $5$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 2^{5} - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{10 {(9 (\sqrt[5]{x} + 2))}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify the numerator.

Apply the product rule to $9 (\sqrt[5]{x} + 2)$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{10 \cdot (9^{4} {(\sqrt[5]{x} + 2)}^{4})}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{10 \cdot (6561 {(\sqrt[5]{x} + 2)}^{4})}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $10$ by $6561$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{65610 {(\sqrt[5]{x} + 2)}^{4}}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor of $65610$ and $6561$ .

Factor $6561$ out of $65610 {(\sqrt[5]{x} + 2)}^{4}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{6561 (10 {(\sqrt[5]{x} + 2)}^{4})}{6561} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factors.

Factor $6561$ out of $6561$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{6561 (10 {(\sqrt[5]{x} + 2)}^{4})}{6561 (1)} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{6561 (10 {(\sqrt[5]{x} + 2)}^{4})}{6561 \cdot 1} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite the expression.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - \frac{10 {(\sqrt[5]{x} + 2)}^{4}}{1} + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Divide $10 {(\sqrt[5]{x} + 2)}^{4}$ by $1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 {(\sqrt[5]{x} + 2)}^{4}) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Use the Binomial Theorem.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 ({\sqrt[5]{x}}^{4} + 4 {\sqrt[5]{x}}^{3} \cdot 2 + 6 {\sqrt[5]{x}}^{2} \cdot 2^{2} + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify each term.

Rewrite ${\sqrt[5]{x}}^{4}$ as ${(x^{4})}^{\frac{1}{5}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 4 {\sqrt[5]{x}}^{3} \cdot 2 + 6 {\sqrt[5]{x}}^{2} \cdot 2^{2} + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 4 \sqrt[5]{x^{3}} \cdot 2 + 6 {\sqrt[5]{x}}^{2} \cdot 2^{2} + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $2$ by $4$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 6 {\sqrt[5]{x}}^{2} \cdot 2^{2} + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite ${\sqrt[5]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{5}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} \cdot 2^{2} + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} \cdot 4 + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $4$ by $6$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 24 \sqrt[5]{x^{2}} + 4 \sqrt[5]{x} \cdot 2^{3} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 24 \sqrt[5]{x^{2}} + 4 \sqrt[5]{x} \cdot 8 + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $8$ by $4$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 24 \sqrt[5]{x^{2}} + 32 \sqrt[5]{x} + 2^{4})) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Raise $2$ to the power of $4$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 (\sqrt[5]{x^{4}} + 8 \sqrt[5]{x^{3}} + 24 \sqrt[5]{x^{2}} + 32 \sqrt[5]{x} + 16)) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 \sqrt[5]{x^{4}} + 10 (8 \sqrt[5]{x^{3}}) + 10 (24 \sqrt[5]{x^{2}}) + 10 (32 \sqrt[5]{x}) + 10 \cdot 16) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify.

Multiply $8$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 \sqrt[5]{x^{4}} + 80 \sqrt[5]{x^{3}} + 10 (24 \sqrt[5]{x^{2}}) + 10 (32 \sqrt[5]{x}) + 10 \cdot 16) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $24$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 \sqrt[5]{x^{4}} + 80 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 10 (32 \sqrt[5]{x}) + 10 \cdot 16) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $32$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 \sqrt[5]{x^{4}} + 80 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 320 \sqrt[5]{x} + 10 \cdot 16) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $10$ by $16$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 \sqrt[5]{x^{4}} + 80 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 320 \sqrt[5]{x} + 160) + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - (10 \sqrt[5]{x^{4}}) - (80 \sqrt[5]{x^{3}}) - (240 \sqrt[5]{x^{2}}) - (320 \sqrt[5]{x}) - 1 \cdot 160 + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify.

Multiply $10$ by $- 1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - (80 \sqrt[5]{x^{3}}) - (240 \sqrt[5]{x^{2}}) - (320 \sqrt[5]{x}) - 1 \cdot 160 + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $80$ by $- 1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - (240 \sqrt[5]{x^{2}}) - (320 \sqrt[5]{x}) - 1 \cdot 160 + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $240$ by $- 1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - (320 \sqrt[5]{x}) - 1 \cdot 160 + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $320$ by $- 1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 1 \cdot 160 + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $- 1$ by $160$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{40 {(9 (\sqrt[5]{x} + 2))}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify the numerator.

Apply the product rule to $9 (\sqrt[5]{x} + 2)$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{40 \cdot (9^{3} {(\sqrt[5]{x} + 2)}^{3})}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Raise $9$ to the power of $3$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{40 \cdot (729 {(\sqrt[5]{x} + 2)}^{3})}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $40$ by $729$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{29160 {(\sqrt[5]{x} + 2)}^{3}}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor of $29160$ and $729$ .

Factor $729$ out of $29160 {(\sqrt[5]{x} + 2)}^{3}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{729 (40 {(\sqrt[5]{x} + 2)}^{3})}{729} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factors.

Factor $729$ out of $729$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{729 (40 {(\sqrt[5]{x} + 2)}^{3})}{729 (1)} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{729 (40 {(\sqrt[5]{x} + 2)}^{3})}{729 \cdot 1} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite the expression.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + \frac{40 {(\sqrt[5]{x} + 2)}^{3}}{1} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Divide $40 {(\sqrt[5]{x} + 2)}^{3}$ by $1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 {(\sqrt[5]{x} + 2)}^{3} - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Use the Binomial Theorem.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 ({\sqrt[5]{x}}^{3} + 3 {\sqrt[5]{x}}^{2} \cdot 2 + 3 \sqrt[5]{x} \cdot 2^{2} + 2^{3}) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify each term.

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 (\sqrt[5]{x^{3}} + 3 {\sqrt[5]{x}}^{2} \cdot 2 + 3 \sqrt[5]{x} \cdot 2^{2} + 2^{3}) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite ${\sqrt[5]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{5}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 (\sqrt[5]{x^{3}} + 3 \sqrt[5]{x^{2}} \cdot 2 + 3 \sqrt[5]{x} \cdot 2^{2} + 2^{3}) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $2$ by $3$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 (\sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} + 3 \sqrt[5]{x} \cdot 2^{2} + 2^{3}) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 (\sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} + 3 \sqrt[5]{x} \cdot 4 + 2^{3}) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $4$ by $3$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 (\sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} + 12 \sqrt[5]{x} + 2^{3}) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 (\sqrt[5]{x^{3}} + 6 \sqrt[5]{x^{2}} + 12 \sqrt[5]{x} + 8) - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 40 (6 \sqrt[5]{x^{2}}) + 40 (12 \sqrt[5]{x}) + 40 \cdot 8 - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify.

Multiply $6$ by $40$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 40 (12 \sqrt[5]{x}) + 40 \cdot 8 - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $12$ by $40$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 40 \cdot 8 - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $40$ by $8$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{80 {(9 (\sqrt[5]{x} + 2))}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify the numerator.

Apply the product rule to $9 (\sqrt[5]{x} + 2)$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{80 \cdot (9^{2} {(\sqrt[5]{x} + 2)}^{2})}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Raise $9$ to the power of $2$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{80 \cdot (81 {(\sqrt[5]{x} + 2)}^{2})}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $80$ by $81$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{6480 {(\sqrt[5]{x} + 2)}^{2}}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor of $6480$ and $81$ .

Factor $81$ out of $6480 {(\sqrt[5]{x} + 2)}^{2}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{81 (80 {(\sqrt[5]{x} + 2)}^{2})}{81} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factors.

Factor $81$ out of $81$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{81 (80 {(\sqrt[5]{x} + 2)}^{2})}{81 (1)} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{81 (80 {(\sqrt[5]{x} + 2)}^{2})}{81 \cdot 1} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite the expression.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - \frac{80 {(\sqrt[5]{x} + 2)}^{2}}{1} + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Divide $80 {(\sqrt[5]{x} + 2)}^{2}$ by $1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 {(\sqrt[5]{x} + 2)}^{2}) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 ((\sqrt[5]{x} + 2) (\sqrt[5]{x} + 2))) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x} (\sqrt[5]{x} + 2) + 2 (\sqrt[5]{x} + 2))) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot 2 + 2 (\sqrt[5]{x} + 2))) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot 2 + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt[5]{x} \sqrt[5]{x}$ .

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot 2 + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x} \sqrt[5]{x} + \sqrt[5]{x} \cdot 2 + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

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

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 ({\sqrt[5]{x}}^{1 + 1} + \sqrt[5]{x} \cdot 2 + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Add $1$ and $1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 ({\sqrt[5]{x}}^{2} + \sqrt[5]{x} \cdot 2 + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Rewrite ${\sqrt[5]{x}}^{2}$ as ${(x^{2})}^{\frac{1}{5}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x^{2}} + \sqrt[5]{x} \cdot 2 + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Move $2$ to the left of $\sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x^{2}} + 2 \cdot \sqrt[5]{x} + 2 \sqrt[5]{x} + 2 \cdot 2)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $2$ by $2$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x^{2}} + 2 \sqrt[5]{x} + 2 \sqrt[5]{x} + 4)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Add $2 \sqrt[5]{x}$ and $2 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 (\sqrt[5]{x^{2}} + 4 \sqrt[5]{x} + 4)) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 \sqrt[5]{x^{2}} + 80 (4 \sqrt[5]{x}) + 80 \cdot 4) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify.

Multiply $4$ by $80$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 \sqrt[5]{x^{2}} + 320 \sqrt[5]{x} + 80 \cdot 4) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $80$ by $4$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 \sqrt[5]{x^{2}} + 320 \sqrt[5]{x} + 320) + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - (80 \sqrt[5]{x^{2}}) - (320 \sqrt[5]{x}) - 1 \cdot 320 + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Simplify.

Multiply $80$ by $- 1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - (320 \sqrt[5]{x}) - 1 \cdot 320 + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $320$ by $- 1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 1 \cdot 320 + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Multiply $- 1$ by $320$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Cancel the common factor of $9$ .

Cancel the common factor.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + \frac{80 (9 (\sqrt[5]{x} + 2))}{9} - 32$

Divide $80 (\sqrt[5]{x} + 2)$ by $1$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 (\sqrt[5]{x} + 2) - 32$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 80 \cdot 2 - 32$

Multiply $80$ by $2$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Simplify by adding terms.

Combine the opposite terms in $x + 10 \sqrt[5]{x^{4}} + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 10 \sqrt[5]{x^{4}} - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$ .

Subtract $10 \sqrt[5]{x^{4}}$ from $10 \sqrt[5]{x^{4}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 0 + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Add $x$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 240 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 240 \sqrt[5]{x^{2}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Add $- 240 \sqrt[5]{x^{2}}$ and $240 \sqrt[5]{x^{2}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} + 0 - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Add $x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x^{2}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} + 320 - 80 \sqrt[5]{x^{2}} - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Subtract $80 \sqrt[5]{x^{2}}$ from $80 \sqrt[5]{x^{2}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 0 + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} + 320 - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Add $x + 40 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} + 320 - 320 \sqrt[5]{x} - 320 + 80 \sqrt[5]{x} + 160 - 32$

Subtract $320$ from $320$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} + 0 - 320 \sqrt[5]{x} + 80 \sqrt[5]{x} + 160 - 32$

Add $x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} - 160 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x} + 160 - 32$

Add $- 160$ and $160$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} + 0 + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x} - 32$

Add $x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 32 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x} - 32$

Subtract $32$ from $32$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} + 0 - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Add $x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} - 80 \sqrt[5]{x^{3}} - 320 \sqrt[5]{x} + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Subtract $80 \sqrt[5]{x^{3}}$ from $40 \sqrt[5]{x^{3}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x - 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} - 320 \sqrt[5]{x} + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Combine the opposite terms in $x - 40 \sqrt[5]{x^{3}} + 80 \sqrt[5]{x} - 320 \sqrt[5]{x} + 40 \sqrt[5]{x^{3}} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$ .

Add $- 40 \sqrt[5]{x^{3}}$ and $40 \sqrt[5]{x^{3}}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 0 + 80 \sqrt[5]{x} - 320 \sqrt[5]{x} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Add $x$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 80 \sqrt[5]{x} - 320 \sqrt[5]{x} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Subtract $320 \sqrt[5]{x}$ from $80 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x - 240 \sqrt[5]{x} + 480 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Add $- 240 \sqrt[5]{x}$ and $480 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 240 \sqrt[5]{x} - 320 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Subtract $320 \sqrt[5]{x}$ from $240 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x - 80 \sqrt[5]{x} + 80 \sqrt[5]{x}$

Combine the opposite terms in $x - 80 \sqrt[5]{x} + 80 \sqrt[5]{x}$ .

Add $- 80 \sqrt[5]{x}$ and $80 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} + 2)) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Remove parentheses.

Simplify each term.

To write $- \frac{10 x^{4}}{6561}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} \cdot \frac{9}{9} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Write each expression with a common denominator of $59049$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{10 x^{4}}{6561}$ by $\frac{9}{9}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5}}{59049} - \frac{10 x^{4} \cdot 9}{6561 \cdot 9} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Multiply $6561$ by $9$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5}}{59049} - \frac{10 x^{4} \cdot 9}{59049} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Combine the numerators over the common denominator.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 10 x^{4} \cdot 9}{59049} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Simplify the numerator.

Factor $x^{4}$ out of $x^{5} - 10 x^{4} \cdot 9$ .

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} x - 10 x^{4} \cdot 9}{59049} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Factor $x^{4}$ out of $- 10 x^{4} \cdot 9$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} x + x^{4} (- 10 \cdot 9)}{59049} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Factor $x^{4}$ out of $x^{4} x + x^{4} (- 10 \cdot 9)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} (x - 10 \cdot 9)}{59049} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Multiply $- 10$ by $9$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} (x - 90)}{59049} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

To write $\frac{40 x^{3}}{729}$ as a fraction with a common denominator, multiply by $\frac{81}{81}$ .

Write each expression with a common denominator of $59049$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{40 x^{3}}{729}$ by $\frac{81}{81}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} (x - 90)}{59049} + \frac{40 x^{3} \cdot 81}{729 \cdot 81} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Multiply $729$ by $81$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} (x - 90)}{59049} + \frac{40 x^{3} \cdot 81}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Combine the numerators over the common denominator.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{4} (x - 90) + 40 x^{3} \cdot 81}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Simplify the numerator.

Factor $x^{3}$ out of $x^{4} (x - 90) + 40 x^{3} \cdot 81$ .

Factor $x^{3}$ out of $x^{4} (x - 90)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x (x - 90)) + 40 x^{3} \cdot 81}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Factor $x^{3}$ out of $40 x^{3} \cdot 81$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x (x - 90)) + x^{3} (40 \cdot 81)}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Factor $x^{3}$ out of $x^{3} (x (x - 90)) + x^{3} (40 \cdot 81)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x (x - 90) + 40 \cdot 81)}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Apply the distributive property.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x \cdot x + x \cdot - 90 + 40 \cdot 81)}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Multiply $x$ by $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x^{2} + x \cdot - 90 + 40 \cdot 81)}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Move $- 90$ to the left of $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x^{2} - 90 \cdot x + 40 \cdot 81)}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

Multiply $40$ by $81$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x^{2} - 90 x + 3240)}{59049} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32} + 2)$

To write $- \frac{80 x^{2}}{81}$ as a fraction with a common denominator, multiply by $\frac{729}{729}$ .

Write each expression with a common denominator of $59049$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{80 x^{2}}{81}$ by $\frac{729}{729}$ .

Multiply $81$ by $729$ .

Combine the numerators over the common denominator.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{3} (x^{2} - 90 x + 3240) - 80 x^{2} \cdot 729}{59049} + \frac{80 x}{9} - 32} + 2)$

Simplify the numerator.

Factor $x^{2}$ out of $x^{3} (x^{2} - 90 x + 3240) - 80 x^{2} \cdot 729$ .

Factor $x^{2}$ out of $x^{3} (x^{2} - 90 x + 3240)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x (x^{2} - 90 x + 3240)) - 80 x^{2} \cdot 729}{59049} + \frac{80 x}{9} - 32} + 2)$

Factor $x^{2}$ out of $- 80 x^{2} \cdot 729$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x (x^{2} - 90 x + 3240)) + x^{2} (- 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Factor $x^{2}$ out of $x^{2} (x (x^{2} - 90 x + 3240)) + x^{2} (- 80 \cdot 729)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x (x^{2} - 90 x + 3240) - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Apply the distributive property.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x \cdot x^{2} + x (- 90 x) + x \cdot 3240 - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Simplify.

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

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

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

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{1 + 2} + x (- 90 x) + x \cdot 3240 - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Add $1$ and $2$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} + x (- 90 x) + x \cdot 3240 - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} - 90 x \cdot x + x \cdot 3240 - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Move $3240$ to the left of $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} - 90 x \cdot x + 3240 \cdot x - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} - 90 (x \cdot x) + 3240 \cdot x - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Multiply $x$ by $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} - 90 x^{2} + 3240 \cdot x - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} - 90 x^{2} + 3240 x - 80 \cdot 729)}{59049} + \frac{80 x}{9} - 32} + 2)$

Multiply $- 80$ by $729$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{2} (x^{3} - 90 x^{2} + 3240 x - 58320)}{59049} + \frac{80 x}{9} - 32} + 2)$

To write $\frac{80 x}{9}$ as a fraction with a common denominator, multiply by $\frac{6561}{6561}$ .

Write each expression with a common denominator of $59049$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{80 x}{9}$ by $\frac{6561}{6561}$ .

Multiply $9$ by $6561$ .

Combine the numerators over the common denominator.

Simplify the numerator.

Factor $x$ out of $x^{2} (x^{3} - 90 x^{2} + 3240 x - 58320) + 80 x \cdot 6561$ .

Factor $x$ out of $x^{2} (x^{3} - 90 x^{2} + 3240 x - 58320)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x (x^{3} - 90 x^{2} + 3240 x - 58320)) + 80 x \cdot 6561}{59049} - 32} + 2)$

Factor $x$ out of $80 x \cdot 6561$ .

Factor $x$ out of $x (x (x^{3} - 90 x^{2} + 3240 x - 58320)) + x (80 \cdot 6561)$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x (x^{3} - 90 x^{2} + 3240 x - 58320) + 80 \cdot 6561)}{59049} - 32} + 2)$

Apply the distributive property.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x \cdot x^{3} + x (- 90 x^{2}) + x (3240 x) + x \cdot - 58320 + 80 \cdot 6561)}{59049} - 32} + 2)$

Simplify.

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

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

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

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{1 + 3} + x (- 90 x^{2}) + x (3240 x) + x \cdot - 58320 + 80 \cdot 6561)}{59049} - 32} + 2)$

Add $1$ and $3$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} + x (- 90 x^{2}) + x (3240 x) + x \cdot - 58320 + 80 \cdot 6561)}{59049} - 32} + 2)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x \cdot x^{2} + x (3240 x) + x \cdot - 58320 + 80 \cdot 6561)}{59049} - 32} + 2)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x \cdot x^{2} + 3240 x \cdot x + x \cdot - 58320 + 80 \cdot 6561)}{59049} - 32} + 2)$

Move $- 58320$ to the left of $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x \cdot x^{2} + 3240 x \cdot x - 58320 \cdot x + 80 \cdot 6561)}{59049} - 32} + 2)$

Simplify each term.

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

Move $x^{2}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 (x^{2} x) + 3240 x \cdot x - 58320 \cdot x + 80 \cdot 6561)}{59049} - 32} + 2)$

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

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

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x^{2 + 1} + 3240 x \cdot x - 58320 \cdot x + 80 \cdot 6561)}{59049} - 32} + 2)$

Add $2$ and $1$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x^{3} + 3240 x \cdot x - 58320 \cdot x + 80 \cdot 6561)}{59049} - 32} + 2)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x^{3} + 3240 (x \cdot x) - 58320 \cdot x + 80 \cdot 6561)}{59049} - 32} + 2)$

Multiply $x$ by $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x^{3} + 3240 x^{2} - 58320 \cdot x + 80 \cdot 6561)}{59049} - 32} + 2)$

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x (x^{4} - 90 x^{3} + 3240 x^{2} - 58320 x + 80 \cdot 6561)}{59049} - 32} + 2)$

Multiply $80$ by $6561$ .

To write $- 32$ as a fraction with a common denominator, multiply by $\frac{59049}{59049}$ .

Combine $- 32$ and $\frac{59049}{59049}$ .

Combine the numerators over the common denominator.

Simplify the numerator.

Apply the distributive property.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x \cdot x^{4} + x (- 90 x^{3}) + x (3240 x^{2}) + x (- 58320 x) + x \cdot 524880 - 32 \cdot 59049}{59049}} + 2)$

Simplify.

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

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

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

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{1 + 4} + x (- 90 x^{3}) + x (3240 x^{2}) + x (- 58320 x) + x \cdot 524880 - 32 \cdot 59049}{59049}} + 2)$

Add $1$ and $4$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} + x (- 90 x^{3}) + x (3240 x^{2}) + x (- 58320 x) + x \cdot 524880 - 32 \cdot 59049}{59049}} + 2)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x \cdot x^{3} + x (3240 x^{2}) + x (- 58320 x) + x \cdot 524880 - 32 \cdot 59049}{59049}} + 2)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x \cdot x^{3} + 3240 x \cdot x^{2} + x (- 58320 x) + x \cdot 524880 - 32 \cdot 59049}{59049}} + 2)$

Rewrite using the commutative property of multiplication.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x \cdot x^{3} + 3240 x \cdot x^{2} - 58320 x \cdot x + x \cdot 524880 - 32 \cdot 59049}{59049}} + 2)$

Move $524880$ to the left of $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x \cdot x^{3} + 3240 x \cdot x^{2} - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

Simplify each term.

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

Move $x^{3}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 (x^{3} x) + 3240 x \cdot x^{2} - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

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

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

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{3 + 1} + 3240 x \cdot x^{2} - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

Add $3$ and $1$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{4} + 3240 x \cdot x^{2} - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

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

Move $x^{2}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{4} + 3240 (x^{2} x) - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

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

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

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{4} + 3240 x^{2 + 1} - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

Add $2$ and $1$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{4} + 3240 x^{3} - 58320 x \cdot x + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{4} + 3240 x^{3} - 58320 (x \cdot x) + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

Multiply $x$ by $x$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 90 x^{4} + 3240 x^{3} - 58320 x^{2} + 524880 \cdot x - 32 \cdot 59049}{59049}} + 2)$

Multiply $- 32$ by $59049$ .

Rewrite $x^{5} - 90 x^{4} + 3240 x^{3} - 58320 x^{2} + 524880 x - 1889568$ in a factored form.

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{x^{5} - 5 \cdot (18 x^{4}) + 10 \cdot (18^{2} x^{3}) - 10 \cdot (18^{3} x^{2}) + 5 \cdot (18^{4} x) - 1 \cdot 18^{5}}{59049}} + 2)$

Factor $x^{5} - 5 \cdot 18 x^{4} + 10 \cdot 18^{2} x^{3} - 10 \cdot 18^{3} x^{2} + 5 \cdot 18^{4} x - 1 \cdot 18^{5}$ using the binomial theorem.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{{(x - 18)}^{5}}{59049}} + 2)$

Rewrite $59049$ as $9^{5}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{\frac{{(x - 18)}^{5}}{9^{5}}} + 2)$

Rewrite $\frac{{(x - 18)}^{5}}{9^{5}}$ as ${(\frac{x - 18}{9})}^{5}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\sqrt[5]{{(\frac{x - 18}{9})}^{5}} + 2)$

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

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x - 18}{9} + 2)$

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x - 18}{9} + 2 \cdot \frac{9}{9})$

Simplify terms.

Combine $2$ and $\frac{9}{9}$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x - 18}{9} + \frac{2 \cdot 9}{9})$

Combine the numerators over the common denominator.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x - 18 + 2 \cdot 9}{9})$

Simplify the numerator.

Multiply $2$ by $9$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x - 18 + 18}{9})$

Add $- 18$ and $18$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x + 0}{9})$

Add $x$ and $0$ .

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x}{9})$

Cancel the common factor of $9$ .

Cancel the common factor.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = 9 (\frac{x}{9})$

Rewrite the expression.

$f (\frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32$ is the inverse of $f (x) = 9 (\sqrt[5]{x} + 2)$ .

$f^{- 1} (x) = \frac{x^{5}}{59049} - \frac{10 x^{4}}{6561} + \frac{40 x^{3}}{729} - \frac{80 x^{2}}{81} + \frac{80 x}{9} - 32$

Find the Inverse f(x)=9( fifth root of x+2)

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