Find the Inverse f(x)=9( fifth root of x-8)+10

$f (x) = 9 (\sqrt[5]{x} - 8) + 10$

Step 1

Write $f (x) = 9 (\sqrt[5]{x} - 8) + 10$ as an equation.

$y = 9 (\sqrt[5]{x} - 8) + 10$

Step 2

Interchange the variables.

$x = 9 (\sqrt[5]{y} - 8) + 10$

Step 3

Solve for $y$ .

Rewrite the equation as $9 (\sqrt[5]{y} - 8) + 10 = x$ .

$9 (\sqrt[5]{y} - 8) + 10 = x$

Solve for $\sqrt[5]{y}$ .

Simplify $9 (\sqrt[5]{y} - 8) + 10$ .

Simplify each term.

Apply the distributive property.

$9 \sqrt[5]{y} + 9 \cdot - 8 + 10 = x$

Multiply $9$ by $- 8$ .

$9 \sqrt[5]{y} - 72 + 10 = x$

Add $- 72$ and $10$ .

$9 \sqrt[5]{y} - 62 = x$

Add $62$ to both sides of the equation.

$9 \sqrt[5]{y} = x + 62$

Divide each term in $9 \sqrt[5]{y} = x + 62$ by $9$ and simplify.

Divide each term in $9 \sqrt[5]{y} = x + 62$ by $9$ .

$\frac{9 \sqrt[5]{y}}{9} = \frac{x}{9} + \frac{62}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 \sqrt[5]{y}}{9} = \frac{x}{9} + \frac{62}{9}$

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

$\sqrt[5]{y} = \frac{x}{9} + \frac{62}{9}$

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

${\sqrt[5]{y}}^{5} = {(\frac{x}{9} + \frac{62}{9})}^{5}$

Simplify each side of the equation.

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

${(y^{\frac{1}{5}})}^{5} = {(\frac{x}{9} + \frac{62}{9})}^{5}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$y^{\frac{1}{5} \cdot 5} = {(\frac{x}{9} + \frac{62}{9})}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = {(\frac{x}{9} + \frac{62}{9})}^{5}$

Rewrite the expression.

$y^{1} = {(\frac{x}{9} + \frac{62}{9})}^{5}$

Simplify.

$y = {(\frac{x}{9} + \frac{62}{9})}^{5}$

Simplify the right side.

Simplify ${(\frac{x}{9} + \frac{62}{9})}^{5}$ .

Use the Binomial Theorem.

$y = {(\frac{x}{9})}^{5} + 5 {(\frac{x}{9})}^{4} \frac{62}{9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Simplify each term.

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

$y = \frac{x^{5}}{9^{5}} + 5 {(\frac{x}{9})}^{4} \frac{62}{9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + 5 {(\frac{x}{9})}^{4} \frac{62}{9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + 5 \frac{x^{4}}{9^{4}} \frac{62}{9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + 5 \frac{x^{4}}{6561} \frac{62}{9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{5 x^{4}}{6561} \cdot \frac{62}{9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Combine.

$y = \frac{x^{5}}{59049} + \frac{5 x^{4} \cdot 62}{6561 \cdot 9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Multiply $62$ by $5$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{6561 \cdot 9} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Multiply $6561$ by $9$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + 10 {(\frac{x}{9})}^{3} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + 10 \frac{x^{3}}{9^{3}} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + 10 \frac{x^{3}}{729} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{10 x^{3}}{729} {(\frac{62}{9})}^{2} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Apply the product rule to $\frac{62}{9}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{10 x^{3}}{729} \cdot \frac{62^{2}}{9^{2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Combine.

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{10 x^{3} \cdot 62^{2}}{729 \cdot 9^{2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Simplify the numerator.

Raise $62$ to the power of $2$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{10 x^{3} \cdot 3844}{729 \cdot 9^{2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Multiply $3844$ by $10$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{729 \cdot 9^{2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Simplify the denominator.

Rewrite $729$ as $3^{6}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{3^{6} \cdot 9^{2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Rewrite $9$ as $3^{2}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{3^{6} \cdot {(3^{2})}^{2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{3^{6} \cdot 3^{2 \cdot 2}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Multiply $2$ by $2$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{3^{6} \cdot 3^{4}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{3^{6 + 4}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Add $6$ and $4$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{3^{10}} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Raise $3$ to the power of $10$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + 10 {(\frac{x}{9})}^{2} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + 10 \frac{x^{2}}{9^{2}} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + 10 \frac{x^{2}}{81} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{10 x^{2}}{81} {(\frac{62}{9})}^{3} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Apply the product rule to $\frac{62}{9}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{10 x^{2}}{81} \cdot \frac{62^{3}}{9^{3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Combine.

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{10 x^{2} \cdot 62^{3}}{81 \cdot 9^{3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Simplify the numerator.

Raise $62$ to the power of $3$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{10 x^{2} \cdot 238328}{81 \cdot 9^{3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Multiply $238328$ by $10$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{81 \cdot 9^{3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Simplify the denominator.

Rewrite $81$ as $3^{4}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{3^{4} \cdot 9^{3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Rewrite $9$ as $3^{2}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{3^{4} \cdot {(3^{2})}^{3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{3^{4} \cdot 3^{2 \cdot 3}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Multiply $2$ by $3$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{3^{4} \cdot 3^{6}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{3^{4 + 6}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Add $4$ and $6$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{3^{10}} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Raise $3$ to the power of $10$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + 5 \frac{x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x}{9} {(\frac{62}{9})}^{4} + {(\frac{62}{9})}^{5}$

Apply the product rule to $\frac{62}{9}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x}{9} \cdot \frac{62^{4}}{9^{4}} + {(\frac{62}{9})}^{5}$

Combine.

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x \cdot 62^{4}}{9 \cdot 9^{4}} + {(\frac{62}{9})}^{5}$

Multiply $9$ by $9^{4}$ by adding the exponents.

Multiply $9$ by $9^{4}$ .

Raise $9$ to the power of $1$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x \cdot 62^{4}}{9^{1} \cdot 9^{4}} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x \cdot 62^{4}}{9^{1 + 4}} + {(\frac{62}{9})}^{5}$

Add $1$ and $4$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x \cdot 62^{4}}{9^{5}} + {(\frac{62}{9})}^{5}$

Simplify the numerator.

Raise $62$ to the power of $4$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{5 x \cdot 14776336}{9^{5}} + {(\frac{62}{9})}^{5}$

Multiply $14776336$ by $5$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{9^{5}} + {(\frac{62}{9})}^{5}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + {(\frac{62}{9})}^{5}$

Apply the product rule to $\frac{62}{9}$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{62^{5}}{9^{5}}$

Raise $62$ to the power of $5$ .

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{9^{5}}$

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

$y = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}$

Step 4

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

$f^{- 1} (x) = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}$ is the inverse of $f (x) = 9 (\sqrt[5]{x} - 8) + 10$ .

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 $f^{- 1} (9 (\sqrt[5]{x} - 8) + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{{(9 (\sqrt[5]{x} - 8) + 10)}^{5}}{59049} + \frac{310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4}}{59049} + \frac{38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3}}{59049} + \frac{2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2}}{59049} + \frac{73881680 (9 (\sqrt[5]{x} - 8) + 10)}{59049} + \frac{916132832}{59049}$

Combine the numerators over the common denominator.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{{(9 (\sqrt[5]{x} - 8) + 10)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{{(9 \sqrt[5]{x} + 9 \cdot - 8 + 10)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $- 8$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{{(9 \sqrt[5]{x} - 72 + 10)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Add $- 72$ and $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{{(9 \sqrt[5]{x} - 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Use the Binomial Theorem.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{{(9 \sqrt[5]{x})}^{5} + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{9^{5} {\sqrt[5]{x}}^{5} + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 {\sqrt[5]{x}}^{5} + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 {(x^{\frac{1}{5}})}^{5} + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x^{\frac{1}{5} \cdot 5} + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x^{\frac{5}{5}} + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Cancel the common factor of $5$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 5 {(9 \sqrt[5]{x})}^{4} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify.

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 5 (9^{4} {\sqrt[5]{x}}^{4}) \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 5 (6561 {\sqrt[5]{x}}^{4}) \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 5 (6561 \sqrt[5]{x^{4}}) \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $6561$ by $5$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 32805 \sqrt[5]{x^{4}} \cdot - 62 + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 62$ by $32805$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 10 {(9 \sqrt[5]{x})}^{3} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 10 (9^{3} {\sqrt[5]{x}}^{3}) {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 10 (729 {\sqrt[5]{x}}^{3}) {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 10 (729 \sqrt[5]{x^{3}}) {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $729$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 7290 \sqrt[5]{x^{3}} {(- 62)}^{2} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $2$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 7290 \sqrt[5]{x^{3}} \cdot 3844 + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $3844$ by $7290$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} + 10 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} + 10 (9^{2} {\sqrt[5]{x}}^{2}) {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} + 10 (81 {\sqrt[5]{x}}^{2}) {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} + 10 (81 \sqrt[5]{x^{2}}) {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $81$ by $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} + 810 \sqrt[5]{x^{2}} {(- 62)}^{3} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $3$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} + 810 \sqrt[5]{x^{2}} \cdot - 238328 + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 238328$ by $810$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 5 (9 \sqrt[5]{x}) {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $5$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 45 \sqrt[5]{x} {(- 62)}^{4} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $4$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 45 \sqrt[5]{x} \cdot 14776336 + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $14776336$ by $45$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} + {(- 62)}^{5} + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $5$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 {(9 (\sqrt[5]{x} - 8) + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 {(9 \sqrt[5]{x} + 9 \cdot - 8 + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $- 8$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 {(9 \sqrt[5]{x} - 72 + 10)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Add $- 72$ and $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 {(9 \sqrt[5]{x} - 62)}^{4} + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Use the Binomial Theorem.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 ({(9 \sqrt[5]{x})}^{4} + 4 {(9 \sqrt[5]{x})}^{3} \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (9^{4} {\sqrt[5]{x}}^{4} + 4 {(9 \sqrt[5]{x})}^{3} \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 {\sqrt[5]{x}}^{4} + 4 {(9 \sqrt[5]{x})}^{3} \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} + 4 {(9 \sqrt[5]{x})}^{3} \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} + 4 (9^{3} {\sqrt[5]{x}}^{3}) \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} + 4 (729 {\sqrt[5]{x}}^{3}) \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} + 4 (729 \sqrt[5]{x^{3}}) \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $729$ by $4$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} + 2916 \sqrt[5]{x^{3}} \cdot - 62 + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 62$ by $2916$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 6 {(9 \sqrt[5]{x})}^{2} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 6 (9^{2} {\sqrt[5]{x}}^{2}) {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 6 (81 {\sqrt[5]{x}}^{2}) {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 6 (81 \sqrt[5]{x^{2}}) {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $81$ by $6$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 486 \sqrt[5]{x^{2}} {(- 62)}^{2} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $2$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 486 \sqrt[5]{x^{2}} \cdot 3844 + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $3844$ by $486$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 1868184 \sqrt[5]{x^{2}} + 4 (9 \sqrt[5]{x}) {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $4$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 1868184 \sqrt[5]{x^{2}} + 36 \sqrt[5]{x} {(- 62)}^{3} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $3$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 1868184 \sqrt[5]{x^{2}} + 36 \sqrt[5]{x} \cdot - 238328 + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 238328$ by $36$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 1868184 \sqrt[5]{x^{2}} - 8579808 \sqrt[5]{x} + {(- 62)}^{4}) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $4$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}} - 180792 \sqrt[5]{x^{3}} + 1868184 \sqrt[5]{x^{2}} - 8579808 \sqrt[5]{x} + 14776336) + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 310 (6561 \sqrt[5]{x^{4}}) + 310 (- 180792 \sqrt[5]{x^{3}}) + 310 (1868184 \sqrt[5]{x^{2}}) + 310 (- 8579808 \sqrt[5]{x}) + 310 \cdot 14776336 + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify.

Multiply $6561$ by $310$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} + 310 (- 180792 \sqrt[5]{x^{3}}) + 310 (1868184 \sqrt[5]{x^{2}}) + 310 (- 8579808 \sqrt[5]{x}) + 310 \cdot 14776336 + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 180792$ by $310$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 310 (1868184 \sqrt[5]{x^{2}}) + 310 (- 8579808 \sqrt[5]{x}) + 310 \cdot 14776336 + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $1868184$ by $310$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} + 310 (- 8579808 \sqrt[5]{x}) + 310 \cdot 14776336 + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 8579808$ by $310$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 310 \cdot 14776336 + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $310$ by $14776336$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 {(9 (\sqrt[5]{x} - 8) + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 {(9 \sqrt[5]{x} + 9 \cdot - 8 + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $- 8$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 {(9 \sqrt[5]{x} - 72 + 10)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Add $- 72$ and $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 {(9 \sqrt[5]{x} - 62)}^{3} + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Use the Binomial Theorem.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 ({(9 \sqrt[5]{x})}^{3} + 3 {(9 \sqrt[5]{x})}^{2} \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (9^{3} {\sqrt[5]{x}}^{3} + 3 {(9 \sqrt[5]{x})}^{2} \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 {\sqrt[5]{x}}^{3} + 3 {(9 \sqrt[5]{x})}^{2} \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} + 3 {(9 \sqrt[5]{x})}^{2} \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} + 3 (9^{2} {\sqrt[5]{x}}^{2}) \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} + 3 (81 {\sqrt[5]{x}}^{2}) \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} + 3 (81 \sqrt[5]{x^{2}}) \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $81$ by $3$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} + 243 \sqrt[5]{x^{2}} \cdot - 62 + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 62$ by $243$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} - 15066 \sqrt[5]{x^{2}} + 3 (9 \sqrt[5]{x}) {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $3$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} - 15066 \sqrt[5]{x^{2}} + 27 \sqrt[5]{x} {(- 62)}^{2} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $2$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} - 15066 \sqrt[5]{x^{2}} + 27 \sqrt[5]{x} \cdot 3844 + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $3844$ by $27$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} - 15066 \sqrt[5]{x^{2}} + 103788 \sqrt[5]{x} + {(- 62)}^{3}) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Raise $- 62$ to the power of $3$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}} - 15066 \sqrt[5]{x^{2}} + 103788 \sqrt[5]{x} - 238328) + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 38440 (729 \sqrt[5]{x^{3}}) + 38440 (- 15066 \sqrt[5]{x^{2}}) + 38440 (103788 \sqrt[5]{x}) + 38440 \cdot - 238328 + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify.

Multiply $729$ by $38440$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 38440 (- 15066 \sqrt[5]{x^{2}}) + 38440 (103788 \sqrt[5]{x}) + 38440 \cdot - 238328 + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 15066$ by $38440$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 38440 (103788 \sqrt[5]{x}) + 38440 \cdot - 238328 + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $103788$ by $38440$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} + 38440 \cdot - 238328 + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $38440$ by $- 238328$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 {(9 (\sqrt[5]{x} - 8) + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 {(9 \sqrt[5]{x} + 9 \cdot - 8 + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $- 8$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 {(9 \sqrt[5]{x} - 72 + 10)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Add $- 72$ and $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 {(9 \sqrt[5]{x} - 62)}^{2} + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Rewrite ${(9 \sqrt[5]{x} - 62)}^{2}$ as $(9 \sqrt[5]{x} - 62) (9 \sqrt[5]{x} - 62)$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 ((9 \sqrt[5]{x} - 62) (9 \sqrt[5]{x} - 62)) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Expand $(9 \sqrt[5]{x} - 62) (9 \sqrt[5]{x} - 62)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (9 \sqrt[5]{x} (9 \sqrt[5]{x} - 62) - 62 (9 \sqrt[5]{x} - 62)) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (9 \sqrt[5]{x} (9 \sqrt[5]{x}) + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x} - 62)) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (9 \sqrt[5]{x} (9 \sqrt[5]{x}) + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify and combine like terms.

Simplify each term.

Multiply $9 \sqrt[5]{x} (9 \sqrt[5]{x})$ .

Multiply $9$ by $9$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x} \sqrt[5]{x} + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 (\sqrt[5]{x} \sqrt[5]{x}) + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 (\sqrt[5]{x} \sqrt[5]{x}) + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 {\sqrt[5]{x}}^{1 + 1} + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Add $1$ and $1$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 {\sqrt[5]{x}}^{2} + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

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

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x^{2}} + 9 \sqrt[5]{x} \cdot - 62 - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 62$ by $9$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x^{2}} - 558 \sqrt[5]{x} - 62 (9 \sqrt[5]{x}) - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $9$ by $- 62$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x^{2}} - 558 \sqrt[5]{x} - 558 \sqrt[5]{x} - 62 \cdot - 62) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 62$ by $- 62$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x^{2}} - 558 \sqrt[5]{x} - 558 \sqrt[5]{x} + 3844) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Subtract $558 \sqrt[5]{x}$ from $- 558 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x^{2}} - 1116 \sqrt[5]{x} + 3844) + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 2383280 (81 \sqrt[5]{x^{2}}) + 2383280 (- 1116 \sqrt[5]{x}) + 2383280 \cdot 3844 + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify.

Multiply $81$ by $2383280$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} + 2383280 (- 1116 \sqrt[5]{x}) + 2383280 \cdot 3844 + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $- 1116$ by $2383280$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 2383280 \cdot 3844 + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Multiply $2383280$ by $3844$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 73881680 (9 (\sqrt[5]{x} - 8) + 10) + 916132832}{59049}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 73881680 (9 \sqrt[5]{x} + 9 \cdot - 8 + 10) + 916132832}{59049}$

Multiply $9$ by $- 8$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 73881680 (9 \sqrt[5]{x} - 72 + 10) + 916132832}{59049}$

Add $- 72$ and $10$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 73881680 (9 \sqrt[5]{x} - 62) + 916132832}{59049}$

Apply the distributive property.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 73881680 (9 \sqrt[5]{x}) + 73881680 \cdot - 62 + 916132832}{59049}$

Multiply $9$ by $73881680$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} + 73881680 \cdot - 62 + 916132832}{59049}$

Multiply $73881680$ by $- 62$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Simplify terms.

Combine the opposite terms in $59049 x - 2033910 \sqrt[5]{x^{4}} + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 + 2033910 \sqrt[5]{x^{4}} - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832$ .

Add $- 2033910 \sqrt[5]{x^{4}}$ and $2033910 \sqrt[5]{x^{4}}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 0 + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Add $59049 x$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} + 579137040 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} - 579137040 \sqrt[5]{x^{2}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Subtract $579137040 \sqrt[5]{x^{2}}$ from $579137040 \sqrt[5]{x^{2}}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} + 0 - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Add $59049 x + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} - 193045680 \sqrt[5]{x^{2}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 9161328320 + 193045680 \sqrt[5]{x^{2}} - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Add $- 193045680 \sqrt[5]{x^{2}}$ and $193045680 \sqrt[5]{x^{2}}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 0 + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 9161328320 - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Add $59049 x + 28022760 \sqrt[5]{x^{3}}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 9161328320 - 2659740480 \sqrt[5]{x} + 9161328320 + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Add $- 9161328320$ and $9161328320$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} + 0 - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Add $59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 4580664160 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x} - 4580664160 + 916132832}{59049}$

Subtract $4580664160$ from $4580664160$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 0 + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x} + 916132832}{59049}$

Add $59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 916132832 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x} + 916132832}{59049}$

Add $- 916132832$ and $916132832$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} + 0 - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Add $59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x}$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 56045520 \sqrt[5]{x^{3}} - 2659740480 \sqrt[5]{x} + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Subtract $56045520 \sqrt[5]{x^{3}}$ from $28022760 \sqrt[5]{x^{3}}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Combine the opposite terms in $59049 x - 28022760 \sqrt[5]{x^{3}} + 664935120 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 28022760 \sqrt[5]{x^{3}} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}$ .

Add $- 28022760 \sqrt[5]{x^{3}}$ and $28022760 \sqrt[5]{x^{3}}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 0 + 664935120 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Add $59049 x$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 664935120 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Subtract $2659740480 \sqrt[5]{x}$ from $664935120 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 1994805360 \sqrt[5]{x} + 3989610720 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Add $- 1994805360 \sqrt[5]{x}$ and $3989610720 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 1994805360 \sqrt[5]{x} - 2659740480 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Subtract $2659740480 \sqrt[5]{x}$ from $1994805360 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x - 664935120 \sqrt[5]{x} + 664935120 \sqrt[5]{x}}{59049}$

Combine the opposite terms in $59049 x - 664935120 \sqrt[5]{x} + 664935120 \sqrt[5]{x}$ .

Add $- 664935120 \sqrt[5]{x}$ and $664935120 \sqrt[5]{x}$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x + 0}{59049}$

Add $59049 x$ and $0$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x}{59049}$

Cancel the common factor of $59049$ .

Cancel the common factor.

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = \frac{59049 x}{59049}$

Divide $x$ by $1$ .

$f^{- 1} (9 (\sqrt[5]{x} - 8) + 10) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\sqrt[5]{\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}} - 8) + 10$

Remove parentheses.

Simplify each term.

Combine the numerators over the common denominator.

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\sqrt[5]{\frac{x^{5} + 310 x^{4} + 38440 x^{3} + 2383280 x^{2} + 73881680 x + 916132832}{59049}} - 8) + 10$

Simplify the numerator.

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

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\sqrt[5]{\frac{x^{5} + 5 \cdot (62 x^{4}) + 10 \cdot (62^{2} x^{3}) + 10 \cdot (62^{3} x^{2}) + 5 \cdot (62^{4} x) + 62^{5}}{59049}} - 8) + 10$

Factor $x^{5} + 5 \cdot 62 x^{4} + 10 \cdot 62^{2} x^{3} + 10 \cdot 62^{3} x^{2} + 5 \cdot 62^{4} x + 62^{5}$ using the binomial theorem.

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

Rewrite $\frac{{(x + 62)}^{5}}{9^{5}}$ as ${(\frac{x + 62}{9})}^{5}$ .

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

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\frac{x + 62}{9} - 8) + 10$

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

Combine $- 8$ and $\frac{9}{9}$ .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply $- 8$ by $9$ .

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\frac{x + 62 - 72}{9}) + 10$

Subtract $72$ from $62$ .

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\frac{x - 10}{9}) + 10$

Cancel the common factor of $9$ .

Cancel the common factor.

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = 9 (\frac{x - 10}{9}) + 10$

Rewrite the expression.

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = x - 10 + 10$

Combine the opposite terms in $x - 10 + 10$ .

Add $- 10$ and $10$ .

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = x + 0$

Add $x$ and $0$ .

$f (\frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}$ is the inverse of $f (x) = 9 (\sqrt[5]{x} - 8) + 10$ .

$f^{- 1} (x) = \frac{x^{5}}{59049} + \frac{310 x^{4}}{59049} + \frac{38440 x^{3}}{59049} + \frac{2383280 x^{2}}{59049} + \frac{73881680 x}{59049} + \frac{916132832}{59049}$

Find the Inverse f(x)=9( fifth root of x-8)+10

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