Algebra Examples

$g (x) = \sqrt[5]{2 x - 1} - 1$

Step 1

Write $g (x) = \sqrt[5]{2 x - 1} - 1$ as an equation.

$y = \sqrt[5]{2 x - 1} - 1$

Step 2

Interchange the variables.

$x = \sqrt[5]{2 y - 1} - 1$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[5]{2 y - 1} - 1 = x$ .

$\sqrt[5]{2 y - 1} - 1 = x$

Step 3.2

Add $1$ to both sides of the equation.

$\sqrt[5]{2 y - 1} = x + 1$

Step 3.3

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

${\sqrt[5]{2 y - 1}}^{5} = {(x + 1)}^{5}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 y - 1}$ as ${(2 y - 1)}^{\frac{1}{5}}$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

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

Step 3.4.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

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

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x + 1)}^{5}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$2 y - 1 = x^{5} + 5 x^{4} \cdot 1 + 10 x^{3} \cdot 1^{2} + 10 x^{2} \cdot 1^{3} + 5 x \cdot 1^{4} + 1^{5}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $5$ by $1$ .

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} \cdot 1^{2} + 10 x^{2} \cdot 1^{3} + 5 x \cdot 1^{4} + 1^{5}$

Step 3.4.3.1.2.2

One to any power is one.

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} \cdot 1 + 10 x^{2} \cdot 1^{3} + 5 x \cdot 1^{4} + 1^{5}$

Step 3.4.3.1.2.3

Multiply $10$ by $1$ .

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} \cdot 1^{3} + 5 x \cdot 1^{4} + 1^{5}$

Step 3.4.3.1.2.4

One to any power is one.

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} \cdot 1 + 5 x \cdot 1^{4} + 1^{5}$

Step 3.4.3.1.2.5

Multiply $10$ by $1$ .

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x \cdot 1^{4} + 1^{5}$

Step 3.4.3.1.2.6

One to any power is one.

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x \cdot 1 + 1^{5}$

Step 3.4.3.1.2.7

Multiply $5$ by $1$ .

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1^{5}$

Step 3.4.3.1.2.8

One to any power is one.

$2 y - 1 = x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1$

Step 3.5

Solve for $y$ .

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Step 3.5.1

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

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Step 3.5.1.1

Add $1$ to both sides of the equation.

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

Step 3.5.1.2

Add $1$ and $1$ .

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

Step 3.5.2

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

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Step 3.5.2.1

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

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

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Simplify each term.

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Step 3.5.2.3.1.1

Cancel the common factor of $10$ and $2$ .

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Step 3.5.2.3.1.1.1

Factor $2$ out of $10 x^{3}$ .

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

Step 3.5.2.3.1.1.2

Cancel the common factors.

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Step 3.5.2.3.1.1.2.1

Factor $2$ out of $2$ .

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

Step 3.5.2.3.1.1.2.2

Cancel the common factor.

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

Step 3.5.2.3.1.1.2.3

Rewrite the expression.

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

Step 3.5.2.3.1.1.2.4

Divide $5 x^{3}$ by $1$ .

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

Step 3.5.2.3.1.2

Cancel the common factor of $10$ and $2$ .

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Step 3.5.2.3.1.2.1

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

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

Step 3.5.2.3.1.2.2

Cancel the common factors.

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Step 3.5.2.3.1.2.2.1

Factor $2$ out of $2$ .

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

Step 3.5.2.3.1.2.2.2

Cancel the common factor.

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

Step 3.5.2.3.1.2.2.3

Rewrite the expression.

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

Step 3.5.2.3.1.2.2.4

Divide $5 x^{2}$ by $1$ .

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

Step 3.5.2.3.1.3

Divide $2$ by $2$ .

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

Step 4

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

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

Step 5

Verify if $g^{- 1} (x) = \frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1$ is the inverse of $g (x) = \sqrt[5]{2 x - 1} - 1$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $g^{- 1} (\sqrt[5]{2 x - 1} - 1)$ by substituting in the value of $g$ into $g^{- 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3

Simplify terms.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Use the Binomial Theorem.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 ({\sqrt[5]{2 x - 1}}^{3} + 3 {\sqrt[5]{2 x - 1}}^{2} \cdot - 1 + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.2

Simplify each term.

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Step 5.2.3.1.2.1

Rewrite ${\sqrt[5]{2 x - 1}}^{3}$ as $\sqrt[5]{{(2 x - 1)}^{3}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 (\sqrt[5]{{(2 x - 1)}^{3}} + 3 {\sqrt[5]{2 x - 1}}^{2} \cdot - 1 + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.2.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 (\sqrt[5]{{(2 x - 1)}^{3}} + 3 \sqrt[5]{{(2 x - 1)}^{2}} \cdot - 1 + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.2.3

Multiply $- 1$ by $3$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.2.4

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} \cdot 1 + {(- 1)}^{3}) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.2.5

Multiply $3$ by $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} + {(- 1)}^{3}) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.2.6

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} - 1) + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.3

Apply the distributive property.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} + 5 (- 3 \sqrt[5]{{(2 x - 1)}^{2}}) + 5 (3 \sqrt[5]{2 x - 1}) + 5 \cdot - 1 + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.4

Simplify.

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Step 5.2.3.1.4.1

Multiply $- 3$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 5 (3 \sqrt[5]{2 x - 1}) + 5 \cdot - 1 + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.4.2

Multiply $3$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \cdot - 1 + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.4.3

Multiply $5$ by $- 1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 {(\sqrt[5]{2 x - 1} - 1)}^{2} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.5

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 ((\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{2 x - 1} - 1)) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.6

Expand $(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{2 x - 1} - 1)$ using the FOIL Method.

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Step 5.2.3.1.6.1

Apply the distributive property.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{2 x - 1} (\sqrt[5]{2 x - 1} - 1) - 1 (\sqrt[5]{2 x - 1} - 1)) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.6.2

Apply the distributive property.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{2 x - 1} \sqrt[5]{2 x - 1} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 (\sqrt[5]{2 x - 1} - 1)) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.6.3

Apply the distributive property.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{2 x - 1} \sqrt[5]{2 x - 1} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7

Simplify and combine like terms.

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Step 5.2.3.1.7.1

Simplify each term.

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Step 5.2.3.1.7.1.1

Multiply $\sqrt[5]{2 x - 1} \sqrt[5]{2 x - 1}$ .

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Step 5.2.3.1.7.1.1.1

Raise $\sqrt[5]{2 x - 1}$ to the power of $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{2 x - 1} \sqrt[5]{2 x - 1} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.1.2

Raise $\sqrt[5]{2 x - 1}$ to the power of $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{2 x - 1} \sqrt[5]{2 x - 1} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.1.3

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 ({\sqrt[5]{2 x - 1}}^{1 + 1} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.1.4

Add $1$ and $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 ({\sqrt[5]{2 x - 1}}^{2} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{{(2 x - 1)}^{2}} + \sqrt[5]{2 x - 1} \cdot - 1 - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.3

Move $- 1$ to the left of $\sqrt[5]{2 x - 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{{(2 x - 1)}^{2}} - 1 \cdot \sqrt[5]{2 x - 1} - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.4

Rewrite $- 1 \sqrt[5]{2 x - 1}$ as $- \sqrt[5]{2 x - 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{{(2 x - 1)}^{2}} - \sqrt[5]{2 x - 1} - 1 \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.5

Rewrite $- 1 \sqrt[5]{2 x - 1}$ as $- \sqrt[5]{2 x - 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{{(2 x - 1)}^{2}} - \sqrt[5]{2 x - 1} - \sqrt[5]{2 x - 1} - 1 \cdot - 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.1.6

Multiply $- 1$ by $- 1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{{(2 x - 1)}^{2}} - \sqrt[5]{2 x - 1} - \sqrt[5]{2 x - 1} + 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.7.2

Subtract $\sqrt[5]{2 x - 1}$ from $- \sqrt[5]{2 x - 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 (\sqrt[5]{{(2 x - 1)}^{2}} - 2 \sqrt[5]{2 x - 1} + 1) + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.8

Apply the distributive property.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 \sqrt[5]{{(2 x - 1)}^{2}} + 5 (- 2 \sqrt[5]{2 x - 1}) + 5 \cdot 1 + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.9

Simplify.

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Step 5.2.3.1.9.1

Multiply $- 2$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 5 \cdot 1 + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.1.9.2

Multiply $5$ by $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 5 + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.2

Simplify terms.

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Step 5.2.3.2.1

Combine the opposite terms in $\frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 5 + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$ .

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Step 5.2.3.2.1.1

Add $- 5$ and $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 0 + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.2.1.2

Add $\frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1}$ and $0$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5}}{2} + \frac{5 {(\sqrt[5]{2 x - 1} - 1)}^{4}}{2} + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + \frac{5 (\sqrt[5]{2 x - 1} - 1)}{2} + 1$

Step 5.2.3.2.2

Combine the numerators over the common denominator.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{{(\sqrt[5]{2 x - 1} - 1)}^{5} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{4} + 5 (\sqrt[5]{2 x - 1} - 1)}{2}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

Factor $\sqrt[5]{2 x - 1} - 1$ out of ${(\sqrt[5]{2 x - 1} - 1)}^{5} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{4} + 5 (\sqrt[5]{2 x - 1} - 1)$ .

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Step 5.2.4.1.1

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

Step 5.2.4.1.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) {(\sqrt[5]{2 x - 1} - 1)}^{4} + (\sqrt[5]{2 x - 1} - 1) (5 {(\sqrt[5]{2 x - 1} - 1)}^{3}) + 5 (\sqrt[5]{2 x - 1} - 1)}{2}$

Step 5.2.4.1.3

Factor $\sqrt[5]{2 x - 1} - 1$ out of $5 (\sqrt[5]{2 x - 1} - 1)$ .

Step 5.2.4.1.4

Factor $\sqrt[5]{2 x - 1} - 1$ out of $(\sqrt[5]{2 x - 1} - 1) {(\sqrt[5]{2 x - 1} - 1)}^{4} + (\sqrt[5]{2 x - 1} - 1) (5 {(\sqrt[5]{2 x - 1} - 1)}^{3})$ .

Step 5.2.4.1.5

Factor $\sqrt[5]{2 x - 1} - 1$ out of $(\sqrt[5]{2 x - 1} - 1) ({(\sqrt[5]{2 x - 1} - 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3}) + (\sqrt[5]{2 x - 1} - 1) \cdot 5$ .

Step 5.2.4.2

Use the Binomial Theorem.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) ({\sqrt[5]{2 x - 1}}^{4} + 4 {\sqrt[5]{2 x - 1}}^{3} \cdot - 1 + 6 {\sqrt[5]{2 x - 1}}^{2} {(- 1)}^{2} + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3

Simplify each term.

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Step 5.2.4.3.1

Rewrite ${\sqrt[5]{2 x - 1}}^{4}$ as $\sqrt[5]{{(2 x - 1)}^{4}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 4 {\sqrt[5]{2 x - 1}}^{3} \cdot - 1 + 6 {\sqrt[5]{2 x - 1}}^{2} {(- 1)}^{2} + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.2

Rewrite ${\sqrt[5]{2 x - 1}}^{3}$ as $\sqrt[5]{{(2 x - 1)}^{3}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 4 \sqrt[5]{{(2 x - 1)}^{3}} \cdot - 1 + 6 {\sqrt[5]{2 x - 1}}^{2} {(- 1)}^{2} + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.3

Multiply $- 1$ by $4$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 {\sqrt[5]{2 x - 1}}^{2} {(- 1)}^{2} + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.4

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} {(- 1)}^{2} + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.5

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} \cdot 1 + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.6

Multiply $6$ by $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} + 4 \sqrt[5]{2 x - 1} {(- 1)}^{3} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.7

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} + 4 \sqrt[5]{2 x - 1} \cdot - 1 + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.8

Multiply $- 1$ by $4$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + {(- 1)}^{4} + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.3.9

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 {(\sqrt[5]{2 x - 1} - 1)}^{3} + 5)}{2}$

Step 5.2.4.4

Use the Binomial Theorem.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 ({\sqrt[5]{2 x - 1}}^{3} + 3 {\sqrt[5]{2 x - 1}}^{2} \cdot - 1 + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5)}{2}$

Step 5.2.4.5

Simplify each term.

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Step 5.2.4.5.1

Rewrite ${\sqrt[5]{2 x - 1}}^{3}$ as $\sqrt[5]{{(2 x - 1)}^{3}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 (\sqrt[5]{{(2 x - 1)}^{3}} + 3 {\sqrt[5]{2 x - 1}}^{2} \cdot - 1 + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5)}{2}$

Step 5.2.4.5.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 (\sqrt[5]{{(2 x - 1)}^{3}} + 3 \sqrt[5]{{(2 x - 1)}^{2}} \cdot - 1 + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5)}{2}$

Step 5.2.4.5.3

Multiply $- 1$ by $3$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} {(- 1)}^{2} + {(- 1)}^{3}) + 5)}{2}$

Step 5.2.4.5.4

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} \cdot 1 + {(- 1)}^{3}) + 5)}{2}$

Step 5.2.4.5.5

Multiply $3$ by $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} + {(- 1)}^{3}) + 5)}{2}$

Step 5.2.4.5.6

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 (\sqrt[5]{{(2 x - 1)}^{3}} - 3 \sqrt[5]{{(2 x - 1)}^{2}} + 3 \sqrt[5]{2 x - 1} - 1) + 5)}{2}$

Step 5.2.4.6

Apply the distributive property.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 \sqrt[5]{{(2 x - 1)}^{3}} + 5 (- 3 \sqrt[5]{{(2 x - 1)}^{2}}) + 5 (3 \sqrt[5]{2 x - 1}) + 5 \cdot - 1 + 5)}{2}$

Step 5.2.4.7

Simplify.

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Step 5.2.4.7.1

Multiply $- 3$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 5 (3 \sqrt[5]{2 x - 1}) + 5 \cdot - 1 + 5)}{2}$

Step 5.2.4.7.2

Multiply $3$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \cdot - 1 + 5)}{2}$

Step 5.2.4.7.3

Multiply $5$ by $- 1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{{(2 x - 1)}^{3}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5)}{2}$

Step 5.2.4.8

Add $- 4 \sqrt[5]{{(2 x - 1)}^{3}}$ and $5 \sqrt[5]{{(2 x - 1)}^{3}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 6 \sqrt[5]{{(2 x - 1)}^{2}} - 4 \sqrt[5]{2 x - 1} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5)}{2}$

Step 5.2.4.9

Subtract $15 \sqrt[5]{{(2 x - 1)}^{2}}$ from $6 \sqrt[5]{{(2 x - 1)}^{2}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 \sqrt[5]{2 x - 1} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 5 + 5)}{2}$

Step 5.2.4.10

Add $- 4 \sqrt[5]{2 x - 1}$ and $15 \sqrt[5]{2 x - 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1} - 5 + 5)}{2}$

Step 5.2.4.11

Subtract $5$ from $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} - 4 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1} + 5)}{2}$

Step 5.2.4.12

Add $- 4$ and $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1 + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2}$

Step 5.2.5

To write $5 \sqrt[5]{{(2 x - 1)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = 5 \sqrt[5]{{(2 x - 1)}^{3}} \cdot \frac{2}{2} + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.6

Simplify terms.

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Step 5.2.6.1

Combine $5 \sqrt[5]{{(2 x - 1)}^{3}}$ and $\frac{2}{2}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 \sqrt[5]{{(2 x - 1)}^{3}} \cdot 2}{2} + \frac{(\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.6.2

Combine the numerators over the common denominator.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 \sqrt[5]{{(2 x - 1)}^{3}} \cdot 2 + (\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7

Simplify each term.

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Step 5.2.7.1

Simplify the numerator.

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Step 5.2.7.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(2 x - 1)}^{3}}$ as ${(2 x - 1)}^{\frac{3}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 {(2 x - 1)}^{\frac{3}{5}} \cdot 2 + (\sqrt[5]{2 x - 1} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 x - 1}$ as ${(2 x - 1)}^{\frac{1}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 {(2 x - 1)}^{\frac{3}{5}} \cdot 2 + ({(2 x - 1)}^{\frac{1}{5}} - 1) (\sqrt[5]{{(2 x - 1)}^{4}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(2 x - 1)}^{4}}$ as ${(2 x - 1)}^{\frac{4}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 {(2 x - 1)}^{\frac{3}{5}} \cdot 2 + ({(2 x - 1)}^{\frac{1}{5}} - 1) ({(2 x - 1)}^{\frac{4}{5}} + 1 + \sqrt[5]{{(2 x - 1)}^{3}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(2 x - 1)}^{3}}$ as ${(2 x - 1)}^{\frac{3}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 {(2 x - 1)}^{\frac{3}{5}} \cdot 2 + ({(2 x - 1)}^{\frac{1}{5}} - 1) ({(2 x - 1)}^{\frac{4}{5}} + 1 + {(2 x - 1)}^{\frac{3}{5}} - 9 \sqrt[5]{{(2 x - 1)}^{2}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.5

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{{(2 x - 1)}^{2}}$ as ${(2 x - 1)}^{\frac{2}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 {(2 x - 1)}^{\frac{3}{5}} \cdot 2 + ({(2 x - 1)}^{\frac{1}{5}} - 1) ({(2 x - 1)}^{\frac{4}{5}} + 1 + {(2 x - 1)}^{\frac{3}{5}} - 9 {(2 x - 1)}^{\frac{2}{5}} + 11 \sqrt[5]{2 x - 1})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{2 x - 1}$ as ${(2 x - 1)}^{\frac{1}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{5 {(2 x - 1)}^{\frac{3}{5}} \cdot 2 + ({(2 x - 1)}^{\frac{1}{5}} - 1) ({(2 x - 1)}^{\frac{4}{5}} + 1 + {(2 x - 1)}^{\frac{3}{5}} - 9 {(2 x - 1)}^{\frac{2}{5}} + 11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.7

Multiply $2$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + ({(2 x - 1)}^{\frac{1}{5}} - 1) ({(2 x - 1)}^{\frac{4}{5}} + 1 + {(2 x - 1)}^{\frac{3}{5}} - 9 {(2 x - 1)}^{\frac{2}{5}} + 11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.8

Expand $({(2 x - 1)}^{\frac{1}{5}} - 1) ({(2 x - 1)}^{\frac{4}{5}} + 1 + {(2 x - 1)}^{\frac{3}{5}} - 9 {(2 x - 1)}^{\frac{2}{5}} + 11 {(2 x - 1)}^{\frac{1}{5}})$ by multiplying each term in the first expression by each term in the second expression.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{4}{5}} + {(2 x - 1)}^{\frac{1}{5}} \cdot 1 + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9

Simplify each term.

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Step 5.2.7.1.9.1

Multiply ${(2 x - 1)}^{\frac{1}{5}}$ by ${(2 x - 1)}^{\frac{4}{5}}$ by adding the exponents.

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Step 5.2.7.1.9.1.1

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5} + \frac{4}{5}} + {(2 x - 1)}^{\frac{1}{5}} \cdot 1 + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.1.2

Combine the numerators over the common denominator.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1 + 4}{5}} + {(2 x - 1)}^{\frac{1}{5}} \cdot 1 + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.1.3

Add $1$ and $4$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{5}{5}} + {(2 x - 1)}^{\frac{1}{5}} \cdot 1 + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.1.4

Divide $5$ by $5$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} \cdot 1 + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.2

Simplify ${(2 x - 1)}^{1}$ .

Step 5.2.7.1.9.3

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.4

Multiply ${(2 x - 1)}^{\frac{1}{5}}$ by ${(2 x - 1)}^{\frac{3}{5}}$ by adding the exponents.

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Step 5.2.7.1.9.4.1

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{1}{5} + \frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.4.2

Combine the numerators over the common denominator.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{1 + 3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.4.3

Add $1$ and $3$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} + {(2 x - 1)}^{\frac{1}{5}} (- 9 {(2 x - 1)}^{\frac{2}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.5

Rewrite using the commutative property of multiplication.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{2}{5}} + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.6

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

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Step 5.2.7.1.9.6.1

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 ({(2 x - 1)}^{\frac{2}{5}} {(2 x - 1)}^{\frac{1}{5}}) + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.6.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{2}{5} + \frac{1}{5}} + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.6.3

Combine the numerators over the common denominator.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{2 + 1}{5}} + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.6.4

Add $2$ and $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + {(2 x - 1)}^{\frac{1}{5}} (11 {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.7

Rewrite using the commutative property of multiplication.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{1}{5}} - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.8

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

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Step 5.2.7.1.9.8.1

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 ({(2 x - 1)}^{\frac{1}{5}} {(2 x - 1)}^{\frac{1}{5}}) - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.8.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{1}{5} + \frac{1}{5}} - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.8.3

Combine the numerators over the common denominator.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{1 + 1}{5}} - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.8.4

Add $1$ and $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - 1 {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.9

Rewrite $- 1 {(2 x - 1)}^{\frac{4}{5}}$ as $- {(2 x - 1)}^{\frac{4}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{4}{5}} - 1 \cdot 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.10

Multiply $- 1$ by $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{4}{5}} - 1 - 1 {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.11

Rewrite $- 1 {(2 x - 1)}^{\frac{3}{5}}$ as $- {(2 x - 1)}^{\frac{3}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{4}{5}} - 1 - {(2 x - 1)}^{\frac{3}{5}} - 1 (- 9 {(2 x - 1)}^{\frac{2}{5}}) - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.12

Multiply $- 9$ by $- 1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{4}{5}} - 1 - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 1 (11 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.9.13

Multiply $11$ by $- 1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{4}{5}} - 1 - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 11 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.10

Combine the opposite terms in $2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + {(2 x - 1)}^{\frac{4}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{4}{5}} - 1 - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 11 {(2 x - 1)}^{\frac{1}{5}}$ .

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Step 5.2.7.1.10.1

Subtract ${(2 x - 1)}^{\frac{4}{5}}$ from ${(2 x - 1)}^{\frac{4}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} + 0 - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - 1 - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 11 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.10.2

Add $2 x - 1 + {(2 x - 1)}^{\frac{1}{5}}$ and $0$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 1 + {(2 x - 1)}^{\frac{1}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - 1 - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 11 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.11

Subtract $1$ from $- 1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 2 + {(2 x - 1)}^{\frac{1}{5}} - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 11 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.12

Subtract $11 {(2 x - 1)}^{\frac{1}{5}}$ from ${(2 x - 1)}^{\frac{1}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 2 - 9 {(2 x - 1)}^{\frac{3}{5}} + 11 {(2 x - 1)}^{\frac{2}{5}} - {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.13

Subtract ${(2 x - 1)}^{\frac{3}{5}}$ from $- 9 {(2 x - 1)}^{\frac{3}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 2 + 11 {(2 x - 1)}^{\frac{2}{5}} - 10 {(2 x - 1)}^{\frac{3}{5}} + 9 {(2 x - 1)}^{\frac{2}{5}} - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.14

Add $11 {(2 x - 1)}^{\frac{2}{5}}$ and $9 {(2 x - 1)}^{\frac{2}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{10 {(2 x - 1)}^{\frac{3}{5}} + 2 x - 2 + 20 {(2 x - 1)}^{\frac{2}{5}} - 10 {(2 x - 1)}^{\frac{3}{5}} - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.15

Subtract $10 {(2 x - 1)}^{\frac{3}{5}}$ from $10 {(2 x - 1)}^{\frac{3}{5}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 x - 2 + 20 {(2 x - 1)}^{\frac{2}{5}} + 0 - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.16

Add $2 x$ and $0$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 x - 2 + 20 {(2 x - 1)}^{\frac{2}{5}} - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.17

Reorder terms.

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 x + 20 {(2 x - 1)}^{\frac{2}{5}} - 2 - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18

Factor $2$ out of $2 x + 20 {(2 x - 1)}^{\frac{2}{5}} - 2 - 10 {(2 x - 1)}^{\frac{1}{5}}$ .

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Step 5.2.7.1.18.1

Factor $2$ out of $2 x$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x) + 20 {(2 x - 1)}^{\frac{2}{5}} - 2 - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18.2

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x) + 2 (10 {(2 x - 1)}^{\frac{2}{5}}) - 2 - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18.3

Factor $2$ out of $- 2$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x) + 2 (10 {(2 x - 1)}^{\frac{2}{5}}) + 2 (- 1) - 10 {(2 x - 1)}^{\frac{1}{5}}}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18.4

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x) + 2 (10 {(2 x - 1)}^{\frac{2}{5}}) + 2 (- 1) + 2 (- 5 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18.5

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x + 10 {(2 x - 1)}^{\frac{2}{5}}) + 2 (- 1) + 2 (- 5 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18.6

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x + 10 {(2 x - 1)}^{\frac{2}{5}} - 1) + 2 (- 5 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.1.18.7

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = \frac{2 (x + 10 {(2 x - 1)}^{\frac{2}{5}} - 1 - 5 {(2 x - 1)}^{\frac{1}{5}})}{2} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.7.2

Cancel the common factor.

Step 5.2.7.3

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

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = x + 10 {(2 x - 1)}^{\frac{2}{5}} - 1 - 5 {(2 x - 1)}^{\frac{1}{5}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$

Step 5.2.8

Simplify by adding terms.

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Step 5.2.8.1

Combine the opposite terms in $x + 10 {(2 x - 1)}^{\frac{2}{5}} - 1 - 5 {(2 x - 1)}^{\frac{1}{5}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1} + 1$ .

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Step 5.2.8.1.1

Add $- 1$ and $1$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = x + 10 {(2 x - 1)}^{\frac{2}{5}} + 0 - 5 {(2 x - 1)}^{\frac{1}{5}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1}$

Step 5.2.8.1.2

Add $x + 10 {(2 x - 1)}^{\frac{2}{5}}$ and $0$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = x + 10 {(2 x - 1)}^{\frac{2}{5}} - 5 {(2 x - 1)}^{\frac{1}{5}} - 15 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} + 5 \sqrt[5]{{(2 x - 1)}^{2}} - 10 \sqrt[5]{2 x - 1}$

Step 5.2.8.2

Add $- 15 \sqrt[5]{{(2 x - 1)}^{2}}$ and $5 \sqrt[5]{{(2 x - 1)}^{2}}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = x + 10 {(2 x - 1)}^{\frac{2}{5}} - 5 {(2 x - 1)}^{\frac{1}{5}} - 10 \sqrt[5]{{(2 x - 1)}^{2}} + 15 \sqrt[5]{2 x - 1} - 10 \sqrt[5]{2 x - 1}$

Step 5.2.8.3

Subtract $10 \sqrt[5]{2 x - 1}$ from $15 \sqrt[5]{2 x - 1}$ .

$g^{- 1} (\sqrt[5]{2 x - 1} - 1) = x + 10 {(2 x - 1)}^{\frac{2}{5}} - 5 {(2 x - 1)}^{\frac{1}{5}} - 10 \sqrt[5]{{(2 x - 1)}^{2}} + 5 \sqrt[5]{2 x - 1}$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

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

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{2 (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) - 1} - 1$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{2 (\frac{x^{5}}{2}) + 2 (\frac{5 x^{4}}{2}) + 2 (5 x^{3}) + 2 (5 x^{2}) + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2

Simplify.

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Step 5.3.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.3.2.1.1

Cancel the common factor.

Step 5.3.3.2.1.2

Rewrite the expression.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 2 (\frac{5 x^{4}}{2}) + 2 (5 x^{3}) + 2 (5 x^{2}) + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.2.1

Cancel the common factor.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 2 (\frac{5 x^{4}}{2}) + 2 (5 x^{3}) + 2 (5 x^{2}) + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.2.2

Rewrite the expression.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 2 (5 x^{3}) + 2 (5 x^{2}) + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.3

Multiply $5$ by $2$ .

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 10 x^{3} + 2 (5 x^{2}) + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.4

Multiply $5$ by $2$ .

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.5

Cancel the common factor of $2$ .

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Step 5.3.3.2.5.1

Cancel the common factor.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 2 (\frac{5 x}{2}) + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.5.2

Rewrite the expression.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 2 \cdot 1 - 1} - 1$

Step 5.3.3.2.6

Multiply $2$ by $1$ .

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 2 - 1} - 1$

Step 5.3.3.3

Subtract $1$ from $2$ .

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1} - 1$

Step 5.3.3.4

Factor $x^{5} + 5 x^{4} + 10 x^{3} + 10 x^{2} + 5 x + 1$ using the binomial theorem.

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = \sqrt[5]{{(x + 1)}^{5}} - 1$

Step 5.3.3.5

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

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

Step 5.3.4

Combine the opposite terms in $x + 1 - 1$ .

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Step 5.3.4.1

Subtract $1$ from $1$ .

$g (\frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

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

Step 5.4

Since $g^{- 1} (g (x)) = x$ and $g (g^{- 1} (x)) = x$ , then $g^{- 1} (x) = \frac{x^{5}}{2} + \frac{5 x^{4}}{2} + 5 x^{3} + 5 x^{2} + \frac{5 x}{2} + 1$ is the inverse of $g (x) = \sqrt[5]{2 x - 1} - 1$ .

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