Find the Inverse f(x)=10x^(1/5)-2

$f (x) = 10 x^{\frac{1}{5}} - 2$

Write $f (x) = 10 x^{\frac{1}{5}} - 2$ as an equation.

$y = 10 x^{\frac{1}{5}} - 2$

Interchange the variables.

$x = 10 y^{\frac{1}{5}} - 2$

Solve for $y$ .

Rewrite the equation as $10 y^{\frac{1}{5}} - 2 = x$ .

$10 y^{\frac{1}{5}} - 2 = x$

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

Add $2$ to both sides of the equation.

$10 y^{\frac{1}{5}} - x = 2$

Add $x$ to both sides of the equation.

$10 y^{\frac{1}{5}} = 2 + x$

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

${(10 y^{\frac{1}{5}})}^{5} = {(2 + x)}^{5}$

Simplify the exponent.

Simplify the left side.

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

Apply the product rule to $10 y^{\frac{1}{5}}$ .

$10^{5} {(y^{\frac{1}{5}})}^{5} = {(2 + x)}^{5}$

Raise $10$ to the power of $5$ .

$100000 {(y^{\frac{1}{5}})}^{5} = {(2 + x)}^{5}$

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

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

$100000 y^{\frac{1}{5} \cdot 5} = {(2 + x)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$100000 y^{\frac{1}{5} \cdot 5} = {(2 + x)}^{5}$

Rewrite the expression.

$100000 y^{1} = {(2 + x)}^{5}$

Simplify.

$100000 y = {(2 + x)}^{5}$

Simplify the right side.

Simplify ${(2 + x)}^{5}$ .

Use the Binomial Theorem.

$100000 y = 2^{5} + 5 \cdot 2^{4} x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

Simplify each term.

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

$100000 y = 32 + 5 \cdot 2^{4} x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

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

$100000 y = 32 + 5 \cdot 16 x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

Multiply $5$ by $16$ .

$100000 y = 32 + 80 x + 10 \cdot 2^{3} x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

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

$100000 y = 32 + 80 x + 10 \cdot 8 x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

Multiply $10$ by $8$ .

$100000 y = 32 + 80 x + 80 x^{2} + 10 \cdot 2^{2} x^{3} + 5 \cdot 2 x^{4} + x^{5}$

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

$100000 y = 32 + 80 x + 80 x^{2} + 10 \cdot 4 x^{3} + 5 \cdot 2 x^{4} + x^{5}$

Multiply $10$ by $4$ .

$100000 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 5 \cdot 2 x^{4} + x^{5}$

Multiply $5$ by $2$ .

$100000 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}$

Divide each term in $100000 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}$ by $100000$ and simplify.

Divide each term in $100000 y = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}$ by $100000$ .

$\frac{100000 y}{100000} = \frac{32}{100000} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Simplify the left side.

Cancel the common factor of $100000$ .

Cancel the common factor.

$\frac{100000 y}{100000} = \frac{32}{100000} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Divide $y$ by $1$ .

$y = \frac{32}{100000} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $32$ and $100000$ .

Factor $32$ out of $32$ .

$y = \frac{32 (1)}{100000} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factors.

Factor $32$ out of $100000$ .

$y = \frac{32 \cdot 1}{32 \cdot 3125} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor.

$y = \frac{32 \cdot 1}{32 \cdot 3125} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Rewrite the expression.

$y = \frac{1}{3125} + \frac{80 x}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor of $80$ and $100000$ .

Factor $80$ out of $80 x$ .

$y = \frac{1}{3125} + \frac{80 (x)}{100000} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factors.

Factor $80$ out of $100000$ .

$y = \frac{1}{3125} + \frac{80 x}{80 \cdot 1250} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor.

$y = \frac{1}{3125} + \frac{80 x}{80 \cdot 1250} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Rewrite the expression.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{80 x^{2}}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor of $80$ and $100000$ .

Factor $80$ out of $80 x^{2}$ .

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{80 (x^{2})}{100000} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factors.

Factor $80$ out of $100000$ .

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{80 x^{2}}{80 \cdot 1250} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{80 x^{2}}{80 \cdot 1250} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Rewrite the expression.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{40 x^{3}}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor of $40$ and $100000$ .

Factor $40$ out of $40 x^{3}$ .

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{40 (x^{3})}{100000} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factors.

Factor $40$ out of $100000$ .

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{40 x^{3}}{40 \cdot 2500} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{40 x^{3}}{40 \cdot 2500} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Rewrite the expression.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{10 x^{4}}{100000} + \frac{x^{5}}{100000}$

Cancel the common factor of $10$ and $100000$ .

Factor $10$ out of $10 x^{4}$ .

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{10 (x^{4})}{100000} + \frac{x^{5}}{100000}$

Cancel the common factors.

Factor $10$ out of $100000$ .

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{10 x^{4}}{10 \cdot 10000} + \frac{x^{5}}{100000}$

Cancel the common factor.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{10 x^{4}}{10 \cdot 10000} + \frac{x^{5}}{100000}$

Rewrite the expression.

$y = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}$

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

$f^{- 1} (x) = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}$

Verify if $f^{- 1} (x) = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}$ is the inverse of $f (x) = 10 x^{\frac{1}{5}} - 2$ .

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 $f^{- 1} (10 x^{\frac{1}{5}} - 2)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{10 x^{\frac{1}{5}} - 2}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Simplify terms.

Simplify each term.

Factor $2$ out of $10 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{2 (5 x^{\frac{1}{5}}) - 2}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $- 2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{2 (5 x^{\frac{1}{5}}) + 2 (- 1)}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $2 (5 x^{\frac{1}{5}}) + 2 (- 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{2 (5 x^{\frac{1}{5}} - 1)}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factors.

Factor $2$ out of $1250$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{2 (5 x^{\frac{1}{5}} - 1)}{2 (625)} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{2 (5 x^{\frac{1}{5}} - 1)}{2 \cdot 625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Simplify the numerator.

Factor $2$ out of $10 x^{\frac{1}{5}} - 2$ .

Factor $2$ out of $10 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) - 2)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $- 2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) + 2 (- 1))}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $2 (5 x^{\frac{1}{5}}) + 2 (- 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{{(2 (5 x^{\frac{1}{5}} - 1))}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Apply the product rule to $2 (5 x^{\frac{1}{5}} - 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2^{2} {(5 x^{\frac{1}{5}} - 1)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{4 {(5 x^{\frac{1}{5}} - 1)}^{2}}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $4 {(5 x^{\frac{1}{5}} - 1)}^{2}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 (2 {(5 x^{\frac{1}{5}} - 1)}^{2})}{1250} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factors.

Factor $2$ out of $1250$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 (2 {(5 x^{\frac{1}{5}} - 1)}^{2})}{2 (625)} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 (2 {(5 x^{\frac{1}{5}} - 1)}^{2})}{2 \cdot 625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Simplify the numerator.

Factor $2$ out of $10 x^{\frac{1}{5}} - 2$ .

Factor $2$ out of $10 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) - 2)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $- 2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) + 2 (- 1))}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $2 (5 x^{\frac{1}{5}}) + 2 (- 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{{(2 (5 x^{\frac{1}{5}} - 1))}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Apply the product rule to $2 (5 x^{\frac{1}{5}} - 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2^{3} {(5 x^{\frac{1}{5}} - 1)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{8 {(5 x^{\frac{1}{5}} - 1)}^{3}}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $4$ out of $8 {(5 x^{\frac{1}{5}} - 1)}^{3}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{4 (2 {(5 x^{\frac{1}{5}} - 1)}^{3})}{2500} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factors.

Factor $4$ out of $2500$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{4 (2 {(5 x^{\frac{1}{5}} - 1)}^{3})}{4 (625)} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{4 (2 {(5 x^{\frac{1}{5}} - 1)}^{3})}{4 \cdot 625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Simplify the numerator.

Factor $2$ out of $10 x^{\frac{1}{5}} - 2$ .

Factor $2$ out of $10 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) - 2)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $- 2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) + 2 (- 1))}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $2$ out of $2 (5 x^{\frac{1}{5}}) + 2 (- 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(2 (5 x^{\frac{1}{5}} - 1))}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Apply the product rule to $2 (5 x^{\frac{1}{5}} - 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{2^{4} {(5 x^{\frac{1}{5}} - 1)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{16 {(5 x^{\frac{1}{5}} - 1)}^{4}}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Factor $16$ out of $16 {(5 x^{\frac{1}{5}} - 1)}^{4}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{16 ({(5 x^{\frac{1}{5}} - 1)}^{4})}{10000} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factors.

Factor $16$ out of $10000$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{16 {(5 x^{\frac{1}{5}} - 1)}^{4}}{16 \cdot 625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{16 {(5 x^{\frac{1}{5}} - 1)}^{4}}{16 \cdot 625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{{(10 x^{\frac{1}{5}} - 2)}^{5}}{100000}$

Simplify the numerator.

Factor $2$ out of $10 x^{\frac{1}{5}} - 2$ .

Factor $2$ out of $10 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) - 2)}^{5}}{100000}$

Factor $2$ out of $- 2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{{(2 (5 x^{\frac{1}{5}}) + 2 (- 1))}^{5}}{100000}$

Factor $2$ out of $2 (5 x^{\frac{1}{5}}) + 2 (- 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{{(2 (5 x^{\frac{1}{5}} - 1))}^{5}}{100000}$

Apply the product rule to $2 (5 x^{\frac{1}{5}} - 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{2^{5} {(5 x^{\frac{1}{5}} - 1)}^{5}}{100000}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{32 {(5 x^{\frac{1}{5}} - 1)}^{5}}{100000}$

Factor $32$ out of $32 {(5 x^{\frac{1}{5}} - 1)}^{5}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{32 ({(5 x^{\frac{1}{5}} - 1)}^{5})}{100000}$

Cancel the common factors.

Factor $32$ out of $100000$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{32 {(5 x^{\frac{1}{5}} - 1)}^{5}}{32 \cdot 3125}$

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{32 {(5 x^{\frac{1}{5}} - 1)}^{5}}{32 \cdot 3125}$

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{5 x^{\frac{1}{5}} - 1}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{2}}{625} + \frac{2 {(5 x^{\frac{1}{5}} - 1)}^{3}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{4}}{625} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 {(5 x^{\frac{1}{5}} - 1)}^{2} + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Simplify each term.

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 ((5 x^{\frac{1}{5}} - 1) (5 x^{\frac{1}{5}} - 1)) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

Apply the distributive property.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}} - 1) - 1 (5 x^{\frac{1}{5}} - 1)) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}} - 1)) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 x^{\frac{1}{5}} (5 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 \cdot (5 x^{\frac{1}{5}} x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 \cdot (5 (x^{\frac{1}{5}} x^{\frac{1}{5}})) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 \cdot (5 x^{\frac{1}{5} + \frac{1}{5}}) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 \cdot (5 x^{\frac{1 + 1}{5}}) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Add $1$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (5 \cdot (5 x^{\frac{2}{5}}) + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $5$ by $5$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (25 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} \cdot - 1 - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $- 1$ by $5$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (25 x^{\frac{2}{5}} - 5 x^{\frac{1}{5}} - 1 (5 x^{\frac{1}{5}}) - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $5$ by $- 1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (25 x^{\frac{2}{5}} - 5 x^{\frac{1}{5}} - 5 x^{\frac{1}{5}} - 1 \cdot - 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $- 1$ by $- 1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (25 x^{\frac{2}{5}} - 5 x^{\frac{1}{5}} - 5 x^{\frac{1}{5}} + 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (25 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} + 1) + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 2 (25 x^{\frac{2}{5}}) + 2 (- 10 x^{\frac{1}{5}}) + 2 \cdot 1 + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Simplify.

Multiply $25$ by $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} + 2 (- 10 x^{\frac{1}{5}}) + 2 \cdot 1 + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $- 10$ by $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 \cdot 1 + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $2$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 {(5 x^{\frac{1}{5}} - 1)}^{3} + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Use the Binomial Theorem.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 ({(5 x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Simplify each term.

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (5^{3} {(x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 {(x^{\frac{1}{5}})}^{3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{1}{5} \cdot 3} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} + 3 {(5 x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} + 3 (5^{2} {(x^{\frac{1}{5}})}^{2}) \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} + 3 (25 {(x^{\frac{1}{5}})}^{2}) \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} + 3 (25 x^{\frac{1}{5} \cdot 2}) \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} + 3 (25 x^{\frac{2}{5}}) \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $25$ by $3$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} + 75 x^{\frac{2}{5}} \cdot - 1 + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $- 1$ by $75$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} - 75 x^{\frac{2}{5}} + 3 (5 x^{\frac{1}{5}}) {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $5$ by $3$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} - 75 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Raise $- 1$ to the power of $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} - 75 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} \cdot 1 + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $15$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} - 75 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + {(- 1)}^{3}) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}} - 75 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} - 1) + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Apply the distributive property.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 2 (125 x^{\frac{3}{5}}) + 2 (- 75 x^{\frac{2}{5}}) + 2 (15 x^{\frac{1}{5}}) + 2 \cdot - 1 + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Simplify.

Multiply $125$ by $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} + 2 (- 75 x^{\frac{2}{5}}) + 2 (15 x^{\frac{1}{5}}) + 2 \cdot - 1 + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $- 75$ by $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 2 (15 x^{\frac{1}{5}}) + 2 \cdot - 1 + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $15$ by $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 2 \cdot - 1 + {(5 x^{\frac{1}{5}} - 1)}^{4}}{625}$

Multiply $2$ by $- 1$ .

Use the Binomial Theorem.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + {(5 x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Simplify each term.

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 5^{4} {(x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 {(x^{\frac{1}{5}})}^{4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{1}{5} \cdot 4} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} + 4 {(5 x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} + 4 (5^{3} {(x^{\frac{1}{5}})}^{3}) \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} + 4 (125 {(x^{\frac{1}{5}})}^{3}) \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} + 4 (125 x^{\frac{1}{5} \cdot 3}) \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} + 4 (125 x^{\frac{3}{5}}) \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Multiply $125$ by $4$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} + 500 x^{\frac{3}{5}} \cdot - 1 + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Multiply $- 1$ by $500$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 6 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 6 (5^{2} {(x^{\frac{1}{5}})}^{2}) {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 6 (25 {(x^{\frac{1}{5}})}^{2}) {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 6 (25 x^{\frac{1}{5} \cdot 2}) {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 6 (25 x^{\frac{2}{5}}) {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Multiply $25$ by $6$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} {(- 1)}^{2} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Raise $- 1$ to the power of $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} \cdot 1 + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Multiply $150$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} + 4 (5 x^{\frac{1}{5}}) {(- 1)}^{3} + {(- 1)}^{4}}{625}$

Multiply $5$ by $4$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} + 20 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} + 20 x^{\frac{1}{5}} \cdot - 1 + {(- 1)}^{4}}{625}$

Multiply $- 1$ by $20$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + {(- 1)}^{4}}{625}$

Raise $- 1$ to the power of $4$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 1}{625}$

Simplify by adding terms.

Combine the opposite terms in $5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 2 + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 2 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 1$ .

Subtract $2$ from $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 0 + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 1}{625}$

Add $5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} - 150 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 150 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 1}{625}$

Add $- 150 x^{\frac{2}{5}}$ and $150 x^{\frac{2}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} + 0 - 20 x^{\frac{1}{5}} + 1}{625}$

Add $5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} - 1 + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + 1}{625}$

Add $- 1$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + 0}{625}$

Add $5 x^{\frac{1}{5}} + 50 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}}$ and $0$ .

Subtract $20 x^{\frac{1}{5}}$ from $5 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{50 x^{\frac{2}{5}} - 15 x^{\frac{1}{5}} + 250 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}}}{625}$

Add $- 15 x^{\frac{1}{5}}$ and $30 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{50 x^{\frac{2}{5}} + 250 x^{\frac{3}{5}} + 15 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 500 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}}}{625}$

Subtract $500 x^{\frac{3}{5}}$ from $250 x^{\frac{3}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{50 x^{\frac{2}{5}} + 15 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 250 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}}}{625}$

Subtract $20 x^{\frac{1}{5}}$ from $15 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{50 x^{\frac{2}{5}} - 5 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 250 x^{\frac{3}{5}}}{625}$

Simplify each term.

Simplify the numerator.

Factor $5 x^{\frac{1}{5}}$ out of $50 x^{\frac{2}{5}} - 5 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 250 x^{\frac{3}{5}}$ .

Factor $5 x^{\frac{1}{5}}$ out of $50 x^{\frac{2}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}}) - 5 x^{\frac{1}{5}} + 625 x^{\frac{4}{5}} - 250 x^{\frac{3}{5}}}{625}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} (- 1) + 625 x^{\frac{4}{5}} - 250 x^{\frac{3}{5}}}{625}$

Factor $5 x^{\frac{1}{5}}$ out of $625 x^{\frac{4}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} (- 1) + 5 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) - 250 x^{\frac{3}{5}}}{625}$

Factor $5 x^{\frac{1}{5}}$ out of $- 250 x^{\frac{3}{5}}$ .

Factor $5 x^{\frac{1}{5}}$ out of $5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}}) + 5 x^{\frac{1}{5}} (- 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}} - 1) + 5 x^{\frac{1}{5}} (125 x^{\frac{3}{5}}) + 5 x^{\frac{1}{5}} (- 50 x^{\frac{2}{5}})}{625}$

Factor $5 x^{\frac{1}{5}}$ out of $5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}} - 1) + 5 x^{\frac{1}{5}} (125 x^{\frac{3}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}} - 1 + 125 x^{\frac{3}{5}}) + 5 x^{\frac{1}{5}} (- 50 x^{\frac{2}{5}})}{625}$

Factor $5 x^{\frac{1}{5}}$ out of $5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}} - 1 + 125 x^{\frac{3}{5}}) + 5 x^{\frac{1}{5}} (- 50 x^{\frac{2}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (10 x^{\frac{1}{5}} - 1 + 125 x^{\frac{3}{5}} - 50 x^{\frac{2}{5}})}{625}$

Reorder terms.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{625}$

Factor $5$ out of $5 x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 (x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1))}{625}$

Cancel the common factors.

Factor $5$ out of $625$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{5 (x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1))}{5 \cdot 125}$

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1}{3125} + \frac{{(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + {(5 x^{\frac{1}{5}} - 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify each term.

Simplify the numerator.

Use the Binomial Theorem.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + {(5 x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify each term.

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 5^{5} {(x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 {(x^{\frac{1}{5}})}^{5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x^{\frac{1}{5} \cdot 5} + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Cancel the common factor of $5$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 5 {(5 x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify.

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 5 (5^{4} {(x^{\frac{1}{5}})}^{4}) \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

Move $5^{4}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 5^{4} \cdot (5 {(x^{\frac{1}{5}})}^{4}) \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 5^{4 + 1} {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Add $4$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 5^{5} {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 3125 {(x^{\frac{1}{5}})}^{4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 3125 x^{\frac{1}{5} \cdot 4} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x + 3125 x^{\frac{4}{5}} \cdot - 1 + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $- 1$ by $3125$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 10 {(5 x^{\frac{1}{5}})}^{3} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 10 (5^{3} {(x^{\frac{1}{5}})}^{3}) {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 10 (125 {(x^{\frac{1}{5}})}^{3}) {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 10 (125 x^{\frac{1}{5} \cdot 3}) {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 10 (125 x^{\frac{3}{5}}) {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $125$ by $10$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} {(- 1)}^{2} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Raise $- 1$ to the power of $2$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} \cdot 1 + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $1250$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 10 {(5 x^{\frac{1}{5}})}^{2} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 10 (5^{2} {(x^{\frac{1}{5}})}^{2}) {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 10 (25 {(x^{\frac{1}{5}})}^{2}) {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 10 (25 x^{\frac{1}{5} \cdot 2}) {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 10 (25 x^{\frac{2}{5}}) {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $25$ by $10$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 250 x^{\frac{2}{5}} {(- 1)}^{3} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} + 250 x^{\frac{2}{5}} \cdot - 1 + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $- 1$ by $250$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 5 (5 x^{\frac{1}{5}}) {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $5$ by $5$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} {(- 1)}^{4} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Raise $- 1$ to the power of $4$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} \cdot 1 + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply $25$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} + {(- 1)}^{5}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{1 + 3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} - 1}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Subtract $1$ from $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}} + 0}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Add $3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{3125 x - 3125 x^{\frac{4}{5}} + 1250 x^{\frac{3}{5}} - 250 x^{\frac{2}{5}} + 25 x^{\frac{1}{5}}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Reorder terms.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{3125 x + 1250 x^{\frac{3}{5}} + 25 x^{\frac{1}{5}} - 250 x^{\frac{2}{5}} - 3125 x^{\frac{4}{5}}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Rewrite $3125 x + 1250 x^{\frac{3}{5}} + 25 x^{\frac{1}{5}} - 250 x^{\frac{2}{5}} - 3125 x^{\frac{4}{5}}$ in a factored form.

Factor $25 x^{\frac{1}{5}}$ out of $3125 x + 1250 x^{\frac{3}{5}} + 25 x^{\frac{1}{5}} - 250 x^{\frac{2}{5}} - 3125 x^{\frac{4}{5}}$ .

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 1250 x^{\frac{3}{5}} + 25 x^{\frac{1}{5}} - 250 x^{\frac{2}{5}} - 3125 x^{\frac{4}{5}}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $1250 x^{\frac{3}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} - 250 x^{\frac{2}{5}} - 3125 x^{\frac{4}{5}}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (1) - 250 x^{\frac{2}{5}} - 3125 x^{\frac{4}{5}}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $- 250 x^{\frac{2}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (1) + 25 x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) - 3125 x^{\frac{4}{5}}}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $- 3125 x^{\frac{4}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (1) + 25 x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}})}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 25 x^{\frac{1}{5}} (50 x^{\frac{2}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (1) + 25 x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}})}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}}) + 25 x^{\frac{1}{5}} (1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} + 1) + 25 x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}})}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} + 1) + 25 x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} + 1 - 10 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}})}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25 x^{\frac{1}{5}}$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} + 1 - 10 x^{\frac{1}{5}}) + 25 x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} + 1 - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}})}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Reorder terms.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1)}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $25$ out of $25 x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 (x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1))}{3125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Cancel the common factors.

Factor $25$ out of $3125$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{25 (x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1))}{25 \cdot 125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1)}{125} + \frac{x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1) + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify each term.

Apply the distributive property.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify.

Rewrite using the commutative property of multiplication.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{1}{5}} x^{\frac{4}{5}} + x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Rewrite using the commutative property of multiplication.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{1}{5}} x^{\frac{4}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify each term.

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 (x^{\frac{4}{5}} x^{\frac{1}{5}}) + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{4}{5} + \frac{1}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{4 + 1}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Add $4$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x^{\frac{5}{5}} + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Divide $5$ by $5$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{1}{5}} x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify $125 x^{1}$ .

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 (x^{\frac{2}{5}} x^{\frac{1}{5}}) - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{2}{5} + \frac{1}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{2 + 1}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Add $2$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{1}{5}} x^{\frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 (x^{\frac{1}{5}} x^{\frac{1}{5}}) - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{1}{5} + \frac{1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{1 + 1}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Add $1$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 x^{\frac{1}{5}} x^{\frac{3}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 (x^{\frac{3}{5}} x^{\frac{1}{5}}) + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 x^{\frac{3}{5} + \frac{1}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 x^{\frac{3 + 1}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Add $3$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 x^{\frac{4}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Simplify the numerator.

Factor $x^{\frac{1}{5}}$ out of $125 x + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 x^{\frac{4}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)$ .

Factor $x^{\frac{1}{5}}$ out of $125 x$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + 50 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} - 125 x^{\frac{4}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $x^{\frac{1}{5}}$ out of $50 x^{\frac{3}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) - 10 x^{\frac{2}{5}} - 125 x^{\frac{4}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $x^{\frac{1}{5}}$ out of $- 10 x^{\frac{2}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) - 125 x^{\frac{4}{5}} + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $x^{\frac{1}{5}}$ out of $- 125 x^{\frac{4}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} (50 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Multiply by $1$ .

Factor $x^{\frac{1}{5}}$ out of $x^{\frac{1}{5}} (125 x^{\frac{4}{5}}) + x^{\frac{1}{5}} (50 x^{\frac{2}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $x^{\frac{1}{5}}$ out of $x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}}) + x^{\frac{1}{5}} (- 10 x^{\frac{1}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $x^{\frac{1}{5}}$ out of $x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}}) + x^{\frac{1}{5}} (- 125 x^{\frac{3}{5}})$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1 + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Factor $x^{\frac{1}{5}}$ out of $x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}}) + x^{\frac{1}{5}} \cdot 1$ .

Factor $x^{\frac{1}{5}}$ out of $x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1) + x^{\frac{1}{5}} (125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 50 x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1 + 125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 50 x^{\frac{2}{5}} - 1)}{125}$

Subtract $50 x^{\frac{2}{5}}$ from $50 x^{\frac{2}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1 + 125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} + 0 - 1)}{125}$

Add $125 x^{\frac{4}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} - 10 x^{\frac{1}{5}} - 125 x^{\frac{3}{5}} + 1 + 125 x^{\frac{3}{5}} + 10 x^{\frac{1}{5}} - 1)}{125}$

Add $- 10 x^{\frac{1}{5}}$ and $10 x^{\frac{1}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} - 125 x^{\frac{3}{5}} + 1 + 125 x^{\frac{3}{5}} + 0 - 1)}{125}$

Add $125 x^{\frac{4}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} - 125 x^{\frac{3}{5}} + 1 + 125 x^{\frac{3}{5}} - 1)}{125}$

Add $- 125 x^{\frac{3}{5}}$ and $125 x^{\frac{3}{5}}$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 0 + 1 - 1)}{125}$

Add $125 x^{\frac{4}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 1 - 1)}{125}$

Subtract $1$ from $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} (125 x^{\frac{4}{5}} + 0)}{125}$

Add $125 x^{\frac{4}{5}}$ and $0$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{1}{5}} \cdot (125 x^{\frac{4}{5}})}{125}$

Combine exponents.

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

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{4}{5}} x^{\frac{1}{5}} \cdot 125}{125}$

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

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{4}{5} + \frac{1}{5}} \cdot 125}{125}$

Combine the numerators over the common denominator.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{4 + 1}{5}} \cdot 125}{125}$

Add $4$ and $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x^{\frac{5}{5}} \cdot 125}{125}$

Divide $5$ by $5$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x \cdot 125}{125}$

Simplify $x^{1} \cdot 125$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x \cdot 125}{125}$

Cancel the common factor of $125$ .

Cancel the common factor.

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = \frac{x \cdot 125}{125}$

Divide $x$ by $1$ .

$f^{- 1} (10 x^{\frac{1}{5}} - 2) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}) = 10 {(\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000})}^{\frac{1}{5}} - 2$

Move $\frac{1}{3125}$ .

$f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}) = 10 {(\frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000} + \frac{1}{3125})}^{\frac{1}{5}} - 2$

Move $\frac{x}{1250}$ .

$f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}) = 10 {(\frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000} + \frac{x}{1250} + \frac{1}{3125})}^{\frac{1}{5}} - 2$

Move $\frac{x^{2}}{1250}$ .

$f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}) = 10 {(\frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000} + \frac{x^{2}}{1250} + \frac{x}{1250} + \frac{1}{3125})}^{\frac{1}{5}} - 2$

Move $\frac{x^{3}}{2500}$ .

$f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}) = 10 {(\frac{x^{4}}{10000} + \frac{x^{5}}{100000} + \frac{x^{3}}{2500} + \frac{x^{2}}{1250} + \frac{x}{1250} + \frac{1}{3125})}^{\frac{1}{5}} - 2$

Reorder $\frac{x^{4}}{10000}$ and $\frac{x^{5}}{100000}$ .

$f (\frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}) = 10 {(\frac{x^{5}}{100000} + \frac{x^{4}}{10000} + \frac{x^{3}}{2500} + \frac{x^{2}}{1250} + \frac{x}{1250} + \frac{1}{3125})}^{\frac{1}{5}} - 2$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}$ is the inverse of $f (x) = 10 x^{\frac{1}{5}} - 2$ .

$f^{- 1} (x) = \frac{1}{3125} + \frac{x}{1250} + \frac{x^{2}}{1250} + \frac{x^{3}}{2500} + \frac{x^{4}}{10000} + \frac{x^{5}}{100000}$

Find the Inverse f(x)=10x^(1/5)-2

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