Find the Inverse f(x)=7(x^(1/5)-10)

$f (x) = 7 (x^{\frac{1}{5}} - 10)$

Write $f (x) = 7 (x^{\frac{1}{5}} - 10)$ as an equation.

$y = 7 (x^{\frac{1}{5}} - 10)$

Interchange the variables.

$x = 7 (y^{\frac{1}{5}} - 10)$

Solve for $y$ .

Rewrite the equation as $7 (y^{\frac{1}{5}} - 10) = x$ .

$7 (y^{\frac{1}{5}} - 10) = x$

Divide each term in $7 (y^{\frac{1}{5}} - 10) = x$ by $7$ and simplify.

Divide each term in $7 (y^{\frac{1}{5}} - 10) = x$ by $7$ .

$\frac{7 (y^{\frac{1}{5}} - 10)}{7} = \frac{x}{7}$

Simplify the left side.

Cancel the common factor.

$\frac{7 (y^{\frac{1}{5}} - 10)}{7} = \frac{x}{7}$

Divide $y^{\frac{1}{5}} - 10$ by $1$ .

$y^{\frac{1}{5}} - 10 = \frac{x}{7}$

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

Add $10$ to both sides of the equation.

$y^{\frac{1}{5}} - \frac{x}{7} = 10$

Add $\frac{x}{7}$ to both sides of the equation.

$y^{\frac{1}{5}} = 10 + \frac{x}{7}$

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

${(y^{\frac{1}{5}})}^{5} = {(10 + \frac{x}{7})}^{5}$

Simplify the exponent.

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

$y^{\frac{1}{5} \cdot 5} = {(10 + \frac{x}{7})}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

$y^{\frac{1}{5} \cdot 5} = {(10 + \frac{x}{7})}^{5}$

Rewrite the expression.

$y^{1} = {(10 + \frac{x}{7})}^{5}$

Simplify.

$y = {(10 + \frac{x}{7})}^{5}$

Simplify the right side.

Simplify ${(10 + \frac{x}{7})}^{5}$ .

Use the Binomial Theorem.

$y = 10^{5} + 5 \cdot 10^{4} \frac{x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Simplify each term.

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

$y = 100000 + 5 \cdot 10^{4} \frac{x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Multiply $5 \cdot 10^{4} \frac{x}{7}$ .

Combine $5$ and $\frac{x}{7}$ .

$y = 100000 + 10^{4} \frac{5 x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Combine $10^{4}$ and $\frac{5 x}{7}$ .

$y = 100000 + \frac{10^{4} (5 x)}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Simplify the numerator.

Raise $10$ to the power of $4$ .

$y = 100000 + \frac{10000 \cdot 5 x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Multiply $10000$ by $5$ .

$y = 100000 + \frac{50000 x}{7} + 10 \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Multiply $10$ by $10^{3}$ by adding the exponents.

Multiply $10$ by $10^{3}$ .

Raise $10$ to the power of $1$ .

$y = 100000 + \frac{50000 x}{7} + 10^{1} \cdot 10^{3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

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

$y = 100000 + \frac{50000 x}{7} + 10^{1 + 3} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Add $1$ and $3$ .

$y = 100000 + \frac{50000 x}{7} + 10^{4} {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Raise $10$ to the power of $4$ .

$y = 100000 + \frac{50000 x}{7} + 10000 {(\frac{x}{7})}^{2} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Apply the product rule to $\frac{x}{7}$ .

$y = 100000 + \frac{50000 x}{7} + 10000 \frac{x^{2}}{7^{2}} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Raise $7$ to the power of $2$ .

$y = 100000 + \frac{50000 x}{7} + 10000 \frac{x^{2}}{49} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Combine $10000$ and $\frac{x^{2}}{49}$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10 \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Multiply $10$ by $10^{2}$ by adding the exponents.

Multiply $10$ by $10^{2}$ .

Raise $10$ to the power of $1$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10^{1} \cdot 10^{2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

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

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10^{1 + 2} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Add $1$ and $2$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 10^{3} {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

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

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 1000 {(\frac{x}{7})}^{3} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Apply the product rule to $\frac{x}{7}$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 1000 \frac{x^{3}}{7^{3}} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Raise $7$ to the power of $3$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + 1000 \frac{x^{3}}{343} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Combine $1000$ and $\frac{x^{3}}{343}$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 5 \cdot 10 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Multiply $5$ by $10$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 50 {(\frac{x}{7})}^{4} + {(\frac{x}{7})}^{5}$

Apply the product rule to $\frac{x}{7}$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 50 \frac{x^{4}}{7^{4}} + {(\frac{x}{7})}^{5}$

Raise $7$ to the power of $4$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + 50 \frac{x^{4}}{2401} + {(\frac{x}{7})}^{5}$

Combine $50$ and $\frac{x^{4}}{2401}$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + {(\frac{x}{7})}^{5}$

Apply the product rule to $\frac{x}{7}$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{7^{5}}$

Raise $7$ to the power of $5$ .

$y = 100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807}$

Simplify $100000 + \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807}$ .

Move $100000$ .

$y = \frac{50000 x}{7} + \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + 100000$

Move $\frac{50000 x}{7}$ .

$y = \frac{10000 x^{2}}{49} + \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + \frac{50000 x}{7} + 100000$

Move $\frac{10000 x^{2}}{49}$ .

$y = \frac{1000 x^{3}}{343} + \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

Move $\frac{1000 x^{3}}{343}$ .

$y = \frac{50 x^{4}}{2401} + \frac{x^{5}}{16807} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

Reorder $\frac{50 x^{4}}{2401}$ and $\frac{x^{5}}{16807}$ .

$y = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

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

$f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

Verify if $f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$ is the inverse of $f (x) = 7 (x^{\frac{1}{5}} - 10)$ .

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 $f^{- 1} (7 (x^{\frac{1}{5}} - 10))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = \frac{{(7 (x^{\frac{1}{5}} - 10))}^{5}}{16807} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify each term.

Simplify the numerator.

Apply the product rule to $7 (x^{\frac{1}{5}} - 10)$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = \frac{7^{5} {(x^{\frac{1}{5}} - 10)}^{5}}{16807} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $7$ to the power of $5$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = \frac{16807 {(x^{\frac{1}{5}} - 10)}^{5}}{16807} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factor.

Divide ${(x^{\frac{1}{5}} - 10)}^{5}$ by $1$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = {(x^{\frac{1}{5}} - 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Use the Binomial Theorem.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = {(x^{\frac{1}{5}})}^{5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify each term.

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x^{\frac{1}{5} \cdot 5} + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factor of $5$ .

Cancel the common factor.

Rewrite the expression.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 5 {(x^{\frac{1}{5}})}^{4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify.

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 5 x^{\frac{1}{5} \cdot 4} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 5 x^{\frac{4}{5}} \cdot - 10 + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 10$ by $5$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 {(x^{\frac{1}{5}})}^{3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 x^{\frac{1}{5} \cdot 3} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} {(- 10)}^{2} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $- 10$ to the power of $2$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} \cdot 100 + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $100$ by $10$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 x^{\frac{1}{5} \cdot 2} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 x^{\frac{2}{5}} {(- 10)}^{3} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} + 10 x^{\frac{2}{5}} \cdot - 1000 + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 1000$ by $10$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} {(- 10)}^{4} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $- 10$ to the power of $4$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} \cdot 10000 + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $10000$ by $5$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} + {(- 10)}^{5} + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 {(7 (x^{\frac{1}{5}} - 10))}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify the numerator.

Apply the product rule to $7 (x^{\frac{1}{5}} - 10)$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 \cdot (7^{4} {(x^{\frac{1}{5}} - 10)}^{4})}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $7$ to the power of $4$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 \cdot (2401 {(x^{\frac{1}{5}} - 10)}^{4})}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $50$ by $2401$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{120050 {(x^{\frac{1}{5}} - 10)}^{4}}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Factor $2401$ out of $120050 {(x^{\frac{1}{5}} - 10)}^{4}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{2401 (50 {(x^{\frac{1}{5}} - 10)}^{4})}{2401} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factors.

Factor $2401$ out of $2401$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{2401 (50 {(x^{\frac{1}{5}} - 10)}^{4})}{2401 (1)} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factor.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{2401 (50 {(x^{\frac{1}{5}} - 10)}^{4})}{2401 \cdot 1} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Rewrite the expression.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + \frac{50 {(x^{\frac{1}{5}} - 10)}^{4}}{1} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Divide $50 {(x^{\frac{1}{5}} - 10)}^{4}$ by $1$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 {(x^{\frac{1}{5}} - 10)}^{4} + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Use the Binomial Theorem.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 ({(x^{\frac{1}{5}})}^{4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify each term.

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{1}{5} \cdot 4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} + 4 x^{\frac{1}{5} \cdot 3} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} + 4 x^{\frac{3}{5}} \cdot - 10 + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 10$ by $4$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 {(x^{\frac{1}{5}})}^{2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 x^{\frac{1}{5} \cdot 2} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} {(- 10)}^{2} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $- 10$ to the power of $2$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} \cdot 100 + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $100$ by $6$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} {(- 10)}^{3} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} \cdot - 1000 + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 1000$ by $4$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} - 4000 x^{\frac{1}{5}} + {(- 10)}^{4}) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $- 10$ to the power of $4$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 (x^{\frac{4}{5}} - 40 x^{\frac{3}{5}} + 600 x^{\frac{2}{5}} - 4000 x^{\frac{1}{5}} + 10000) + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} + 50 (- 40 x^{\frac{3}{5}}) + 50 (600 x^{\frac{2}{5}}) + 50 (- 4000 x^{\frac{1}{5}}) + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify.

Multiply $- 40$ by $50$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 50 (600 x^{\frac{2}{5}}) + 50 (- 4000 x^{\frac{1}{5}}) + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $600$ by $50$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} + 50 (- 4000 x^{\frac{1}{5}}) + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 4000$ by $50$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 50 \cdot 10000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $50$ by $10000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 {(7 (x^{\frac{1}{5}} - 10))}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify the numerator.

Apply the product rule to $7 (x^{\frac{1}{5}} - 10)$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 \cdot (7^{3} {(x^{\frac{1}{5}} - 10)}^{3})}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $7$ to the power of $3$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 \cdot (343 {(x^{\frac{1}{5}} - 10)}^{3})}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $1000$ by $343$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343000 {(x^{\frac{1}{5}} - 10)}^{3}}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Factor $343$ out of $343000 {(x^{\frac{1}{5}} - 10)}^{3}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343 (1000 {(x^{\frac{1}{5}} - 10)}^{3})}{343} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factors.

Factor $343$ out of $343$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343 (1000 {(x^{\frac{1}{5}} - 10)}^{3})}{343 (1)} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factor.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{343 (1000 {(x^{\frac{1}{5}} - 10)}^{3})}{343 \cdot 1} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Rewrite the expression.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + \frac{1000 {(x^{\frac{1}{5}} - 10)}^{3}}{1} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Divide $1000 {(x^{\frac{1}{5}} - 10)}^{3}$ by $1$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 {(x^{\frac{1}{5}} - 10)}^{3} + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Use the Binomial Theorem.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 ({(x^{\frac{1}{5}})}^{3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify each term.

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{1}{5} \cdot 3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} + 3 x^{\frac{1}{5} \cdot 2} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} + 3 x^{\frac{2}{5}} \cdot - 10 + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 10$ by $3$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} {(- 10)}^{2} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $- 10$ to the power of $2$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} \cdot 100 + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $100$ by $3$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 300 x^{\frac{1}{5}} + {(- 10)}^{3}) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 (x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 300 x^{\frac{1}{5}} - 1000) + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 1000 (- 30 x^{\frac{2}{5}}) + 1000 (300 x^{\frac{1}{5}}) + 1000 \cdot - 1000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify.

Multiply $- 30$ by $1000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 1000 (300 x^{\frac{1}{5}}) + 1000 \cdot - 1000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $300$ by $1000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} + 1000 \cdot - 1000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $1000$ by $- 1000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 {(7 (x^{\frac{1}{5}} - 10))}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify the numerator.

Apply the product rule to $7 (x^{\frac{1}{5}} - 10)$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 \cdot (7^{2} {(x^{\frac{1}{5}} - 10)}^{2})}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Raise $7$ to the power of $2$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 \cdot (49 {(x^{\frac{1}{5}} - 10)}^{2})}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $10000$ by $49$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{490000 {(x^{\frac{1}{5}} - 10)}^{2}}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Factor $49$ out of $490000 {(x^{\frac{1}{5}} - 10)}^{2}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{49 (10000 {(x^{\frac{1}{5}} - 10)}^{2})}{49} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factors.

Factor $49$ out of $49$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{49 (10000 {(x^{\frac{1}{5}} - 10)}^{2})}{49 (1)} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factor.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{49 (10000 {(x^{\frac{1}{5}} - 10)}^{2})}{49 \cdot 1} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Rewrite the expression.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + \frac{10000 {(x^{\frac{1}{5}} - 10)}^{2}}{1} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Divide $10000 {(x^{\frac{1}{5}} - 10)}^{2}$ by $1$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 {(x^{\frac{1}{5}} - 10)}^{2} + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Rewrite ${(x^{\frac{1}{5}} - 10)}^{2}$ as $(x^{\frac{1}{5}} - 10) (x^{\frac{1}{5}} - 10)$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 ((x^{\frac{1}{5}} - 10) (x^{\frac{1}{5}} - 10)) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Expand $(x^{\frac{1}{5}} - 10) (x^{\frac{1}{5}} - 10)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5}} (x^{\frac{1}{5}} - 10) - 10 (x^{\frac{1}{5}} - 10)) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 (x^{\frac{1}{5}} - 10)) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify and combine like terms.

Simplify each term.

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

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

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1}{5} + \frac{1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Combine the numerators over the common denominator.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{1 + 1}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Add $1$ and $1$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} + x^{\frac{1}{5}} \cdot - 10 - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Move $- 10$ to the left of $x^{\frac{1}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} - 10 \cdot x^{\frac{1}{5}} - 10 x^{\frac{1}{5}} - 10 \cdot - 10) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $- 10$ by $- 10$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} - 10 x^{\frac{1}{5}} - 10 x^{\frac{1}{5}} + 100) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Subtract $10 x^{\frac{1}{5}}$ from $- 10 x^{\frac{1}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 (x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 100) + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} + 10000 (- 20 x^{\frac{1}{5}}) + 10000 \cdot 100 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Simplify.

Multiply $- 20$ by $10000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 10000 \cdot 100 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Multiply $10000$ by $100$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Cancel the common factor.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + \frac{50000 (7 (x^{\frac{1}{5}} - 10))}{7} + 100000$

Divide $50000 (x^{\frac{1}{5}} - 10)$ by $1$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 (x^{\frac{1}{5}} - 10) + 100000$

Apply the distributive property.

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} + 50000 \cdot - 10 + 100000$

Multiply $50000$ by $- 10$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Simplify by adding terms.

Combine the opposite terms in $x - 50 x^{\frac{4}{5}} + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 50 x^{\frac{4}{5}} - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$ .

Add $- 50 x^{\frac{4}{5}}$ and $50 x^{\frac{4}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 + 0 - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Add $x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000$ and $0$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} + 30000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} - 30000 x^{\frac{2}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Subtract $30000 x^{\frac{2}{5}}$ from $30000 x^{\frac{2}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 0 + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Add $x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}}$ and $0$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} - 10000 x^{\frac{2}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 10000 x^{\frac{2}{5}} - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Add $- 10000 x^{\frac{2}{5}}$ and $10000 x^{\frac{2}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000 + 0 - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Add $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000$ and $0$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 1000000 - 200000 x^{\frac{1}{5}} + 1000000 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Add $- 1000000$ and $1000000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 0 + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Add $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 500000 + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}} - 500000 + 100000$

Subtract $500000$ from $500000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}} + 0 + 100000$

Add $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 100000 - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ and $0$ .

Add $- 100000$ and $100000$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}} + 0$

Add $x + 1000 x^{\frac{3}{5}} + 50000 x^{\frac{1}{5}} - 2000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ and $0$ .

Subtract $2000 x^{\frac{3}{5}}$ from $1000 x^{\frac{3}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 50000 x^{\frac{1}{5}} - 1000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

Combine the opposite terms in $x + 50000 x^{\frac{1}{5}} - 1000 x^{\frac{3}{5}} - 200000 x^{\frac{1}{5}} + 1000 x^{\frac{3}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ .

Add $- 1000 x^{\frac{3}{5}}$ and $1000 x^{\frac{3}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 50000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 0 + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

Add $x + 50000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}}$ and $0$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 50000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

Subtract $200000 x^{\frac{1}{5}}$ from $50000 x^{\frac{1}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 150000 x^{\frac{1}{5}} + 300000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

Add $- 150000 x^{\frac{1}{5}}$ and $300000 x^{\frac{1}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 150000 x^{\frac{1}{5}} - 200000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

Subtract $200000 x^{\frac{1}{5}}$ from $150000 x^{\frac{1}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x - 50000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$

Combine the opposite terms in $x - 50000 x^{\frac{1}{5}} + 50000 x^{\frac{1}{5}}$ .

Add $- 50000 x^{\frac{1}{5}}$ and $50000 x^{\frac{1}{5}}$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (7 (x^{\frac{1}{5}} - 10)) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000) = 7 ({(\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000)}^{\frac{1}{5}} - 10)$

Apply the distributive property.

$f (\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000) = 7 {(\frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000)}^{\frac{1}{5}} + 7 \cdot - 10$

Multiply $7$ by $- 10$ .

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$ is the inverse of $f (x) = 7 (x^{\frac{1}{5}} - 10)$ .

$f^{- 1} (x) = \frac{x^{5}}{16807} + \frac{50 x^{4}}{2401} + \frac{1000 x^{3}}{343} + \frac{10000 x^{2}}{49} + \frac{50000 x}{7} + 100000$

Find the Inverse f(x)=7(x^(1/5)-10)

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