Precalculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Interchange the variables.

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

Step 1.3

Solve for $y$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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Step 1.3.2.1

Add $2$ to both sides of the equation.

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

Step 1.3.2.2

Add $x$ to both sides of the equation.

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

Step 1.3.3

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

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

Step 1.3.4

Simplify the exponent.

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Step 1.3.4.1

Simplify the left side.

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Step 1.3.4.1.1

Simplify ${(2 y^{\frac{1}{5}})}^{5}$ .

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Step 1.3.4.1.1.1

Apply the product rule to $2 y^{\frac{1}{5}}$ .

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

Step 1.3.4.1.1.2

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

$32 {(y^{\frac{1}{5}})}^{5} = {(2 + x)}^{5}$

Step 1.3.4.1.1.3

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

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Step 1.3.4.1.1.3.1

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

$32 y^{\frac{1}{5} \cdot 5} = {(2 + x)}^{5}$

Step 1.3.4.1.1.3.2

Cancel the common factor of $5$ .

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Step 1.3.4.1.1.3.2.1

Cancel the common factor.

$32 y^{\frac{1}{5} \cdot 5} = {(2 + x)}^{5}$

Step 1.3.4.1.1.3.2.2

Rewrite the expression.

$32 y^{1} = {(2 + x)}^{5}$

Step 1.3.4.1.1.4

Simplify.

$32 y = {(2 + x)}^{5}$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

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

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Step 1.3.4.2.1.1

Use the Binomial Theorem.

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

Step 1.3.4.2.1.2

Simplify each term.

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Step 1.3.4.2.1.2.1

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

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

Step 1.3.4.2.1.2.2

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

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

Step 1.3.4.2.1.2.3

Multiply $5$ by $16$ .

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

Step 1.3.4.2.1.2.4

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

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

Step 1.3.4.2.1.2.5

Multiply $10$ by $8$ .

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

Step 1.3.4.2.1.2.6

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

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

Step 1.3.4.2.1.2.7

Multiply $10$ by $4$ .

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

Step 1.3.4.2.1.2.8

Multiply $5$ by $2$ .

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

Step 1.3.5

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

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Step 1.3.5.1

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

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

Step 1.3.5.2

Simplify the left side.

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Step 1.3.5.2.1

Cancel the common factor of $32$ .

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Step 1.3.5.2.1.1

Cancel the common factor.

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

Step 1.3.5.2.1.2

Divide $y$ by $1$ .

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

Step 1.3.5.3

Simplify the right side.

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Step 1.3.5.3.1

Simplify each term.

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Step 1.3.5.3.1.1

Divide $32$ by $32$ .

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

Step 1.3.5.3.1.2

Cancel the common factor of $80$ and $32$ .

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Step 1.3.5.3.1.2.1

Factor $16$ out of $80 x$ .

$y = 1 + \frac{16 (5 x)}{32} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.2.2

Cancel the common factors.

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Step 1.3.5.3.1.2.2.1

Factor $16$ out of $32$ .

$y = 1 + \frac{16 (5 x)}{16 (2)} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.2.2.2

Cancel the common factor.

$y = 1 + \frac{16 (5 x)}{16 \cdot 2} + \frac{80 x^{2}}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.2.2.3

Rewrite the expression.

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

Step 1.3.5.3.1.3

Cancel the common factor of $80$ and $32$ .

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Step 1.3.5.3.1.3.1

Factor $16$ out of $80 x^{2}$ .

$y = 1 + \frac{5 x}{2} + \frac{16 (5 x^{2})}{32} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.3.2

Cancel the common factors.

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Step 1.3.5.3.1.3.2.1

Factor $16$ out of $32$ .

$y = 1 + \frac{5 x}{2} + \frac{16 (5 x^{2})}{16 (2)} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.3.2.2

Cancel the common factor.

$y = 1 + \frac{5 x}{2} + \frac{16 (5 x^{2})}{16 \cdot 2} + \frac{40 x^{3}}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.3.2.3

Rewrite the expression.

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

Step 1.3.5.3.1.4

Cancel the common factor of $40$ and $32$ .

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Step 1.3.5.3.1.4.1

Factor $8$ out of $40 x^{3}$ .

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{8 (5 x^{3})}{32} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.4.2

Cancel the common factors.

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Step 1.3.5.3.1.4.2.1

Factor $8$ out of $32$ .

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{8 (5 x^{3})}{8 (4)} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.4.2.2

Cancel the common factor.

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{8 (5 x^{3})}{8 \cdot 4} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.4.2.3

Rewrite the expression.

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{10 x^{4}}{32} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.5

Cancel the common factor of $10$ and $32$ .

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Step 1.3.5.3.1.5.1

Factor $2$ out of $10 x^{4}$ .

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

Step 1.3.5.3.1.5.2

Cancel the common factors.

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Step 1.3.5.3.1.5.2.1

Factor $2$ out of $32$ .

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{2 (5 x^{4})}{2 (16)} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.5.2.2

Cancel the common factor.

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{2 (5 x^{4})}{2 \cdot 16} + \frac{x^{5}}{32}$

Step 1.3.5.3.1.5.2.3

Rewrite the expression.

$y = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$

Step 1.4

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

$f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$

Step 1.5

Verify if $f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$ is the inverse of $f (x) = 2 x^{\frac{1}{5}} - 2$ .

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Step 1.5.1

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

Step 1.5.2

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

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Step 1.5.2.1

Set up the composite result function.

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

Step 1.5.2.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{5 (2 x^{\frac{1}{5}} - 2)}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3

Simplify each term.

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Step 1.5.2.3.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{2 (5 (x^{\frac{1}{5}} - 1))}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.2

Cancel the common factors.

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Step 1.5.2.3.2.1

Factor $2$ out of $2$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{2 (5 (x^{\frac{1}{5}} - 1))}{2 (1)} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.2.2

Cancel the common factor.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{2 (5 (x^{\frac{1}{5}} - 1))}{2 \cdot 1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.2.3

Rewrite the expression.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + \frac{5 (x^{\frac{1}{5}} - 1)}{1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.2.4

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 (x^{\frac{1}{5}} - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.3

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 5 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.4

Multiply $5$ by $- 1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.5

Simplify the numerator.

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Step 1.5.2.3.5.1

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

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Step 1.5.2.3.5.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 (x^{\frac{1}{5}}) - 2)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.5.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 (x^{\frac{1}{5}}) + 2 (- 1))}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.5.1.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 {(2 (x^{\frac{1}{5}} - 1))}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.5.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 \cdot (2^{2} {(x^{\frac{1}{5}} - 1)}^{2})}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.5.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{5 \cdot (4 {(x^{\frac{1}{5}} - 1)}^{2})}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.5.4

Multiply $5$ by $4$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{20 {(x^{\frac{1}{5}} - 1)}^{2}}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.6

Factor $2$ out of $20 {(x^{\frac{1}{5}} - 1)}^{2}$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{2 (10 {(x^{\frac{1}{5}} - 1)}^{2})}{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.7

Cancel the common factors.

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Step 1.5.2.3.7.1

Factor $2$ out of $2$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{2 (10 {(x^{\frac{1}{5}} - 1)}^{2})}{2 (1)} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.7.2

Cancel the common factor.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{2 (10 {(x^{\frac{1}{5}} - 1)}^{2})}{2 \cdot 1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.7.3

Rewrite the expression.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + \frac{10 {(x^{\frac{1}{5}} - 1)}^{2}}{1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.7.4

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 {(x^{\frac{1}{5}} - 1)}^{2} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.8

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 ((x^{\frac{1}{5}} - 1) (x^{\frac{1}{5}} - 1)) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.9

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

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Step 1.5.2.3.9.1

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5}} (x^{\frac{1}{5}} - 1) - 1 (x^{\frac{1}{5}} - 1)) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.9.2

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 (x^{\frac{1}{5}} - 1)) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.9.3

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5}} x^{\frac{1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10

Simplify and combine like terms.

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Step 1.5.2.3.10.1

Simplify each term.

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Step 1.5.2.3.10.1.1

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

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Step 1.5.2.3.10.1.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1}{5} + \frac{1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.1.1.2

Combine the numerators over the common denominator.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{1 + 1}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.1.1.3

Add $1$ and $1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} + x^{\frac{1}{5}} \cdot - 1 - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.1.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - 1 \cdot x^{\frac{1}{5}} - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.1.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - x^{\frac{1}{5}} - 1 x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.1.4

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - x^{\frac{1}{5}} - x^{\frac{1}{5}} - 1 \cdot - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.1.5

Multiply $- 1$ by $- 1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - x^{\frac{1}{5}} - x^{\frac{1}{5}} + 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.10.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 (x^{\frac{2}{5}} - 2 x^{\frac{1}{5}} + 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.11

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} + 10 (- 2 x^{\frac{1}{5}}) + 10 \cdot 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.12

Simplify.

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Step 1.5.2.3.12.1

Multiply $- 2$ by $10$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 \cdot 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.12.2

Multiply $10$ by $1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.13

Simplify the numerator.

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Step 1.5.2.3.13.1

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

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Step 1.5.2.3.13.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 (x^{\frac{1}{5}}) - 2)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.13.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 (x^{\frac{1}{5}}) + 2 (- 1))}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.13.1.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 {(2 (x^{\frac{1}{5}} - 1))}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.13.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 \cdot (2^{3} {(x^{\frac{1}{5}} - 1)}^{3})}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.13.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{5 \cdot (8 {(x^{\frac{1}{5}} - 1)}^{3})}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.13.4

Multiply $5$ by $8$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{40 {(x^{\frac{1}{5}} - 1)}^{3}}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.14

Factor $4$ out of $40 {(x^{\frac{1}{5}} - 1)}^{3}$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{4 (10 {(x^{\frac{1}{5}} - 1)}^{3})}{4} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.15

Cancel the common factors.

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Step 1.5.2.3.15.1

Factor $4$ out of $4$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{4 (10 {(x^{\frac{1}{5}} - 1)}^{3})}{4 (1)} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.15.2

Cancel the common factor.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{4 (10 {(x^{\frac{1}{5}} - 1)}^{3})}{4 \cdot 1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.15.3

Rewrite the expression.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + \frac{10 {(x^{\frac{1}{5}} - 1)}^{3}}{1} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.15.4

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 {(x^{\frac{1}{5}} - 1)}^{3} + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.16

Use the Binomial Theorem.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 ({(x^{\frac{1}{5}})}^{3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17

Simplify each term.

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Step 1.5.2.3.17.1

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

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Step 1.5.2.3.17.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{1}{5} \cdot 3} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.1.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} + 3 {(x^{\frac{1}{5}})}^{2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.2

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

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Step 1.5.2.3.17.2.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} + 3 x^{\frac{1}{5} \cdot 2} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.2.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} + 3 x^{\frac{2}{5}} \cdot - 1 + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.3

Multiply $- 1$ by $3$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} {(- 1)}^{2} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.4

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} \cdot 1 + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.5

Multiply $3$ by $1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} + {(- 1)}^{3}) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.17.6

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 (x^{\frac{3}{5}} - 3 x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} - 1) + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.18

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} + 10 (- 3 x^{\frac{2}{5}}) + 10 (3 x^{\frac{1}{5}}) + 10 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.19

Simplify.

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Step 1.5.2.3.19.1

Multiply $- 3$ by $10$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 10 (3 x^{\frac{1}{5}}) + 10 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.19.2

Multiply $3$ by $10$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} + 10 \cdot - 1 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.19.3

Multiply $10$ by $- 1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 x^{\frac{1}{5}} - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.20

Simplify the numerator.

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Step 1.5.2.3.20.1

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

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Step 1.5.2.3.20.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 (x^{\frac{1}{5}}) - 2)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.20.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 (x^{\frac{1}{5}}) + 2 (- 1))}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.20.1.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(2 (x^{\frac{1}{5}} - 1))}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.20.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 \cdot (2^{4} {(x^{\frac{1}{5}} - 1)}^{4})}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.20.3

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 \cdot (16 {(x^{\frac{1}{5}} - 1)}^{4})}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.20.4

Multiply $5$ by $16$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{80 {(x^{\frac{1}{5}} - 1)}^{4}}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.21

Factor $16$ out of $80 {(x^{\frac{1}{5}} - 1)}^{4}$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{16 (5 {(x^{\frac{1}{5}} - 1)}^{4})}{16} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.22

Cancel the common factors.

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Step 1.5.2.3.22.1

Factor $16$ out of $16$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{16 (5 {(x^{\frac{1}{5}} - 1)}^{4})}{16 (1)} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.22.2

Cancel the common factor.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{16 (5 {(x^{\frac{1}{5}} - 1)}^{4})}{16 \cdot 1} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.22.3

Rewrite the expression.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + \frac{5 {(x^{\frac{1}{5}} - 1)}^{4}}{1} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.22.4

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 {(x^{\frac{1}{5}} - 1)}^{4} + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.23

Use the Binomial Theorem.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 ({(x^{\frac{1}{5}})}^{4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24

Simplify each term.

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Step 1.5.2.3.24.1

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

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Step 1.5.2.3.24.1.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{1}{5} \cdot 4} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.1.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} + 4 {(x^{\frac{1}{5}})}^{3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.2

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

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Step 1.5.2.3.24.2.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} + 4 x^{\frac{1}{5} \cdot 3} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.2.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} + 4 x^{\frac{3}{5}} \cdot - 1 + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.3

Multiply $- 1$ by $4$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 {(x^{\frac{1}{5}})}^{2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.4

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

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Step 1.5.2.3.24.4.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{1}{5} \cdot 2} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.4.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} {(- 1)}^{2} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.5

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} \cdot 1 + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.6

Multiply $6$ by $1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} {(- 1)}^{3} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.7

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} + 4 x^{\frac{1}{5}} \cdot - 1 + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.8

Multiply $- 1$ by $4$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} - 4 x^{\frac{1}{5}} + {(- 1)}^{4}) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.24.9

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 (x^{\frac{4}{5}} - 4 x^{\frac{3}{5}} + 6 x^{\frac{2}{5}} - 4 x^{\frac{1}{5}} + 1) + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.25

Apply the distributive property.

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} + 5 (- 4 x^{\frac{3}{5}}) + 5 (6 x^{\frac{2}{5}}) + 5 (- 4 x^{\frac{1}{5}}) + 5 \cdot 1 + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.26

Simplify.

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Step 1.5.2.3.26.1

Multiply $- 4$ by $5$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 5 (6 x^{\frac{2}{5}}) + 5 (- 4 x^{\frac{1}{5}}) + 5 \cdot 1 + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.26.2

Multiply $6$ by $5$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} + 5 (- 4 x^{\frac{1}{5}}) + 5 \cdot 1 + \frac{{(2 x^{\frac{1}{5}} - 2)}^{5}}{32}$

Step 1.5.2.3.26.3

Multiply $- 4$ by $5$ .

Step 1.5.2.3.26.4

Multiply $5$ by $1$ .

Step 1.5.2.3.27

Simplify the numerator.

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Step 1.5.2.3.27.1

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

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Step 1.5.2.3.27.1.1

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

Step 1.5.2.3.27.1.2

Factor $2$ out of $- 2$ .

Step 1.5.2.3.27.1.3

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

Step 1.5.2.3.27.2

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

Step 1.5.2.3.27.3

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

Step 1.5.2.3.28

Cancel the common factor.

Step 1.5.2.3.29

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

Step 1.5.2.3.30

Use the Binomial Theorem.

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

Step 1.5.2.3.31

Simplify each term.

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Step 1.5.2.3.31.1

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

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Step 1.5.2.3.31.1.1

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

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

Step 1.5.2.3.31.1.2

Cancel the common factor of $5$ .

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Step 1.5.2.3.31.1.2.1

Cancel the common factor.

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

Step 1.5.2.3.31.1.2.2

Rewrite the expression.

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

Step 1.5.2.3.31.2

Simplify.

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

Step 1.5.2.3.31.3

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

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Step 1.5.2.3.31.3.1

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

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

Step 1.5.2.3.31.3.2

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

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

Step 1.5.2.3.31.4

Multiply $- 1$ by $5$ .

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

Step 1.5.2.3.31.5

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

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Step 1.5.2.3.31.5.1

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

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

Step 1.5.2.3.31.5.2

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

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

Step 1.5.2.3.31.6

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

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

Step 1.5.2.3.31.7

Multiply $10$ by $1$ .

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

Step 1.5.2.3.31.8

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

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Step 1.5.2.3.31.8.1

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

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

Step 1.5.2.3.31.8.2

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

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

Step 1.5.2.3.31.9

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

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

Step 1.5.2.3.31.10

Multiply $- 1$ by $10$ .

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

Step 1.5.2.3.31.11

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

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

Step 1.5.2.3.31.12

Multiply $5$ by $1$ .

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

Step 1.5.2.3.31.13

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

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

Step 1.5.2.4

Simplify by adding terms.

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Step 1.5.2.4.1

Combine the opposite terms in $1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}} - 10 + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} + 30 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 5 + x - 5 x^{\frac{4}{5}} + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$ .

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Step 1.5.2.4.1.1

Subtract $10$ from $10$ .

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

Step 1.5.2.4.1.2

Add $1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 30 x^{\frac{2}{5}} + 30 x^{\frac{1}{5}}$ and $0$ .

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

Step 1.5.2.4.1.3

Add $- 30 x^{\frac{2}{5}}$ and $30 x^{\frac{2}{5}}$ .

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

Step 1.5.2.4.1.4

Add $1 + 5 x^{\frac{1}{5}} - 5 + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}}$ and $0$ .

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

Step 1.5.2.4.1.5

Add $- 5$ and $5$ .

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

Step 1.5.2.4.1.6

Add $1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} + 5 x^{\frac{4}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}}$ and $0$ .

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

Step 1.5.2.4.1.7

Subtract $5 x^{\frac{4}{5}}$ from $5 x^{\frac{4}{5}}$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 0 + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

Step 1.5.2.4.1.8

Add $1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x$ and $0$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} + 10 x^{\frac{2}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} - 10 x^{\frac{2}{5}} + 5 x^{\frac{1}{5}} - 1$

Step 1.5.2.4.1.9

Subtract $10 x^{\frac{2}{5}}$ from $10 x^{\frac{2}{5}}$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 1 + 5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 0 + 5 x^{\frac{1}{5}} - 1$

Step 1.5.2.4.1.10

Add $1 + 5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}}$ and $0$ .

Step 1.5.2.4.1.11

Subtract $1$ from $1$ .

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}} + 0$

Step 1.5.2.4.1.12

Add $5 x^{\frac{1}{5}} - 20 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$ and $0$ .

Step 1.5.2.4.2

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = - 15 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} + 30 x^{\frac{1}{5}} - 20 x^{\frac{3}{5}} - 20 x^{\frac{1}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$

Step 1.5.2.4.3

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

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

Step 1.5.2.4.4

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

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

Step 1.5.2.4.5

Combine the opposite terms in $- 5 x^{\frac{1}{5}} + 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}} + 5 x^{\frac{1}{5}}$ .

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Step 1.5.2.4.5.1

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

$f^{- 1} (2 x^{\frac{1}{5}} - 2) = 10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}} + 0$

Step 1.5.2.4.5.2

Add $10 x^{\frac{3}{5}} - 20 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}}$ and $0$ .

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

Step 1.5.2.4.6

Subtract $20 x^{\frac{3}{5}}$ from $10 x^{\frac{3}{5}}$ .

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

Step 1.5.2.4.7

Combine the opposite terms in $- 10 x^{\frac{3}{5}} + x + 10 x^{\frac{3}{5}}$ .

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Step 1.5.2.4.7.1

Add $- 10 x^{\frac{3}{5}}$ and $10 x^{\frac{3}{5}}$ .

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

Step 1.5.2.4.7.2

Add $x$ and $0$ .

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

Step 1.5.3

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

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Step 1.5.3.1

Set up the composite result function.

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

Step 1.5.3.2

Evaluate $f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32})}^{\frac{1}{5}} - 2$

Step 1.5.3.3

Move $1$ .

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + 1)}^{\frac{1}{5}} - 2$

Step 1.5.3.4

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

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

Step 1.5.3.5

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

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

Step 1.5.3.6

Move $\frac{5 x^{3}}{4}$ .

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

Step 1.5.3.7

Reorder $\frac{5 x^{4}}{16}$ and $\frac{x^{5}}{32}$ .

$f (1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}) = 2 {(\frac{x^{5}}{32} + \frac{5 x^{4}}{16} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1)}^{\frac{1}{5}} - 2$

Step 1.5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$ is the inverse of $f (x) = 2 x^{\frac{1}{5}} - 2$ .

$f^{- 1} (x) = 1 + \frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32}$

Step 2

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Step 2.1

Move $1$ .

$\frac{5 x}{2} + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + 1$

Step 2.2

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

$\frac{5 x^{2}}{2} + \frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x}{2} + 1$

Step 2.3

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

$\frac{5 x^{3}}{4} + \frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1$

Step 2.4

Move $\frac{5 x^{3}}{4}$ .

$\frac{5 x^{4}}{16} + \frac{x^{5}}{32} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1$

Step 2.5

Reorder $\frac{5 x^{4}}{16}$ and $\frac{x^{5}}{32}$ .

$\frac{x^{5}}{32} + \frac{5 x^{4}}{16} + \frac{5 x^{3}}{4} + \frac{5 x^{2}}{2} + \frac{5 x}{2} + 1$