Precalculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

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

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

Step 3.3

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

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Step 3.3.1

Add $10$ to both sides of the equation.

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

Step 3.3.2

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

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

Step 3.4

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

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

Step 3.5

Simplify the exponent.

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Step 3.5.1

Simplify the left side.

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Step 3.5.1.1

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

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Step 3.5.1.1.1

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

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Step 3.5.1.1.1.1

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

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

Step 3.5.1.1.1.2

Cancel the common factor of $5$ .

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Step 3.5.1.1.1.2.1

Cancel the common factor.

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

Step 3.5.1.1.1.2.2

Rewrite the expression.

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

Step 3.5.1.1.2

Simplify.

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

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

Use the Binomial Theorem.

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

Step 3.5.2.1.2

Simplify each term.

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Step 3.5.2.1.2.1

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

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

Step 3.5.2.1.2.2

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

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Step 3.5.2.1.2.2.1

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

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

Step 3.5.2.1.2.2.2

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

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

Step 3.5.2.1.2.3

Simplify the numerator.

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Step 3.5.2.1.2.3.1

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

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

Step 3.5.2.1.2.3.2

Multiply $10000$ by $5$ .

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

Step 3.5.2.1.2.4

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

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Step 3.5.2.1.2.4.1

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

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Step 3.5.2.1.2.4.1.1

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

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

Step 3.5.2.1.2.4.1.2

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

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

Step 3.5.2.1.2.4.2

Add $1$ and $3$ .

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

Step 3.5.2.1.2.5

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

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

Step 3.5.2.1.2.6

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

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

Step 3.5.2.1.2.7

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

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

Step 3.5.2.1.2.8

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

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

Step 3.5.2.1.2.9

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

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Step 3.5.2.1.2.9.1

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

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Step 3.5.2.1.2.9.1.1

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

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

Step 3.5.2.1.2.9.1.2

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

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

Step 3.5.2.1.2.9.2

Add $1$ and $2$ .

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

Step 3.5.2.1.2.10

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

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

Step 3.5.2.1.2.11

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

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

Step 3.5.2.1.2.12

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

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

Step 3.5.2.1.2.13

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

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

Step 3.5.2.1.2.14

Multiply $5$ by $10$ .

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

Step 3.5.2.1.2.15

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

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

Step 3.5.2.1.2.16

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

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

Step 3.5.2.1.2.17

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

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

Step 3.5.2.1.2.18

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

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

Step 3.5.2.1.2.19

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

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

Step 3.6

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

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Step 3.6.1

Move $100000$ .

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

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

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

Step 3.6.5

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

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

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

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

Step 5.2.3.1.2

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

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

Step 5.2.3.2

Cancel the common factor.

Step 5.2.3.3

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

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

Step 5.2.3.4

Use the Binomial Theorem.

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

Step 5.2.3.5

Simplify each term.

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Step 5.2.3.5.1

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

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Step 5.2.3.5.1.1

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

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

Step 5.2.3.5.1.2

Cancel the common factor of $5$ .

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Step 5.2.3.5.1.2.1

Cancel the common factor.

Step 5.2.3.5.1.2.2

Rewrite the expression.

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

Step 5.2.3.5.2

Simplify.

Step 5.2.3.5.3

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

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Step 5.2.3.5.3.1

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

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

Step 5.2.3.5.3.2

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

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

Step 5.2.3.5.4

Multiply $- 10$ by $5$ .

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

Step 5.2.3.5.5

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

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Step 5.2.3.5.5.1

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

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

Step 5.2.3.5.5.2

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

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

Step 5.2.3.5.6

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

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

Step 5.2.3.5.7

Multiply $100$ by $10$ .

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

Step 5.2.3.5.8

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

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Step 5.2.3.5.8.1

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

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

Step 5.2.3.5.8.2

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

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

Step 5.2.3.5.9

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

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

Step 5.2.3.5.10

Multiply $- 1000$ by $10$ .

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

Step 5.2.3.5.11

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

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

Step 5.2.3.5.12

Multiply $10000$ by $5$ .

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

Step 5.2.3.5.13

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

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

Step 5.2.3.6

Simplify the numerator.

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Step 5.2.3.6.1

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

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

Step 5.2.3.6.2

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

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

Step 5.2.3.6.3

Multiply $50$ by $2401$ .

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

Step 5.2.3.7

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

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

Step 5.2.3.8

Cancel the common factors.

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Step 5.2.3.8.1

Factor $2401$ out of $2401$ .

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

Step 5.2.3.8.2

Cancel the common factor.

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

Step 5.2.3.8.3

Rewrite the expression.

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

Step 5.2.3.8.4

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

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

Step 5.2.3.9

Use the Binomial Theorem.

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

Step 5.2.3.10

Simplify each term.

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Step 5.2.3.10.1

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

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Step 5.2.3.10.1.1

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

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

Step 5.2.3.10.1.2

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

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

Step 5.2.3.10.2

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

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Step 5.2.3.10.2.1

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

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

Step 5.2.3.10.2.2

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

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

Step 5.2.3.10.3

Multiply $- 10$ by $4$ .

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

Step 5.2.3.10.4

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

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Step 5.2.3.10.4.1

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

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

Step 5.2.3.10.4.2

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

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

Step 5.2.3.10.5

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

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

Step 5.2.3.10.6

Multiply $100$ by $6$ .

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

Step 5.2.3.10.7

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

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

Step 5.2.3.10.8

Multiply $- 1000$ by $4$ .

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

Step 5.2.3.10.9

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

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

Step 5.2.3.11

Apply the distributive property.

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

Step 5.2.3.12

Simplify.

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Step 5.2.3.12.1

Multiply $- 40$ by $50$ .

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

Step 5.2.3.12.2

Multiply $600$ by $50$ .

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

Step 5.2.3.12.3

Multiply $- 4000$ by $50$ .

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

Step 5.2.3.12.4

Multiply $50$ by $10000$ .

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

Step 5.2.3.13

Simplify the numerator.

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Step 5.2.3.13.1

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

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

Step 5.2.3.13.2

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

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

Step 5.2.3.13.3

Multiply $1000$ by $343$ .

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

Step 5.2.3.14

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

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

Step 5.2.3.15

Cancel the common factors.

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Step 5.2.3.15.1

Factor $343$ out of $343$ .

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

Step 5.2.3.15.2

Cancel the common factor.

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

Step 5.2.3.15.3

Rewrite the expression.

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

Step 5.2.3.15.4

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

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

Step 5.2.3.16

Use the Binomial Theorem.

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

Step 5.2.3.17

Simplify each term.

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Step 5.2.3.17.1

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

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Step 5.2.3.17.1.1

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

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

Step 5.2.3.17.1.2

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

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

Step 5.2.3.17.2

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

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Step 5.2.3.17.2.1

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

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

Step 5.2.3.17.2.2

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

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

Step 5.2.3.17.3

Multiply $- 10$ by $3$ .

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

Step 5.2.3.17.4

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

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

Step 5.2.3.17.5

Multiply $100$ by $3$ .

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

Step 5.2.3.17.6

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

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

Step 5.2.3.18

Apply the distributive property.

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

Step 5.2.3.19

Simplify.

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Step 5.2.3.19.1

Multiply $- 30$ by $1000$ .

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

Step 5.2.3.19.2

Multiply $300$ by $1000$ .

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

Step 5.2.3.19.3

Multiply $1000$ by $- 1000$ .

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

Step 5.2.3.20

Simplify the numerator.

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Step 5.2.3.20.1

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

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

Step 5.2.3.20.2

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

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

Step 5.2.3.20.3

Multiply $10000$ by $49$ .

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

Step 5.2.3.21

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

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

Step 5.2.3.22

Cancel the common factors.

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Step 5.2.3.22.1

Factor $49$ out of $49$ .

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

Step 5.2.3.22.2

Cancel the common factor.

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

Step 5.2.3.22.3

Rewrite the expression.

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

Step 5.2.3.22.4

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

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

Step 5.2.3.23

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

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

Step 5.2.3.24

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

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Step 5.2.3.24.1

Apply the distributive property.

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

Step 5.2.3.24.2

Apply the distributive property.

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

Step 5.2.3.24.3

Apply the distributive property.

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

Step 5.2.3.25

Simplify and combine like terms.

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Step 5.2.3.25.1

Simplify each term.

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Step 5.2.3.25.1.1

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

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Step 5.2.3.25.1.1.1

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

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

Step 5.2.3.25.1.1.2

Combine the numerators over the common denominator.

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

Step 5.2.3.25.1.1.3

Add $1$ and $1$ .

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

Step 5.2.3.25.1.2

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

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

Step 5.2.3.25.1.3

Multiply $- 10$ by $- 10$ .

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

Step 5.2.3.25.2

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

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

Step 5.2.3.26

Apply the distributive property.

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

Step 5.2.3.27

Simplify.

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Step 5.2.3.27.1

Multiply $- 20$ by $10000$ .

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

Step 5.2.3.27.2

Multiply $10000$ by $100$ .

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

Step 5.2.3.28

Cancel the common factor.

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

Step 5.2.3.29

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

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

Step 5.2.3.30

Apply the distributive property.

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

Step 5.2.3.31

Multiply $50000$ by $- 10$ .

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

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

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

Step 5.2.4.1.2

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

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

Step 5.2.4.1.3

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

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

Step 5.2.4.1.4

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 $0$ .

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

Step 5.2.4.1.5

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

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

Step 5.2.4.1.6

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 $0$ .

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

Step 5.2.4.1.7

Add $- 1000000$ and $1000000$ .

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

Step 5.2.4.1.8

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 $0$ .

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

Step 5.2.4.1.9

Subtract $500000$ from $500000$ .

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

Step 5.2.4.1.10

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 $0$ .

Step 5.2.4.1.11

Add $- 100000$ and $100000$ .

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

Step 5.2.4.1.12

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 $0$ .

Step 5.2.4.2

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

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

Step 5.2.4.3

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

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Step 5.2.4.3.1

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

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

Step 5.2.4.3.2

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

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

Step 5.2.4.4

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

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

Step 5.2.4.5

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

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

Step 5.2.4.6

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

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

Step 5.2.4.7

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

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Step 5.2.4.7.1

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

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

Step 5.2.4.7.2

Add $x$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

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

Step 5.3.3

Apply the distributive property.

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

Step 5.3.4

Multiply $7$ by $- 10$ .

Step 5.4

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

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