Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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Step 3.1

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

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

Step 3.2

Add $1$ to both sides of the equation.

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

Step 3.3

Raise each side of the equation to the power of $7$ to eliminate the fractional exponent on the left side.

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

Step 3.4

Simplify the exponent.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify ${({(\frac{y^{5}}{7})}^{\frac{1}{7}})}^{7}$ .

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Step 3.4.1.1.1

Multiply the exponents in ${({(\frac{y^{5}}{7})}^{\frac{1}{7}})}^{7}$ .

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Step 3.4.1.1.1.1

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

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

Step 3.4.1.1.1.2

Cancel the common factor of $7$ .

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Step 3.4.1.1.1.2.1

Cancel the common factor.

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

Step 3.4.1.1.1.2.2

Rewrite the expression.

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

Step 3.4.1.1.2

Simplify.

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

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify ${(x + 1)}^{7}$ .

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Step 3.4.2.1.1

Use the Binomial Theorem.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} \cdot 1 + 21 x^{5} \cdot 1^{2} + 35 x^{4} \cdot 1^{3} + 35 x^{3} \cdot 1^{4} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2

Simplify each term.

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Step 3.4.2.1.2.1

Multiply $7$ by $1$ .

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} \cdot 1^{2} + 35 x^{4} \cdot 1^{3} + 35 x^{3} \cdot 1^{4} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.2

One to any power is one.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} \cdot 1 + 35 x^{4} \cdot 1^{3} + 35 x^{3} \cdot 1^{4} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.3

Multiply $21$ by $1$ .

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} \cdot 1^{3} + 35 x^{3} \cdot 1^{4} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.4

One to any power is one.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} \cdot 1 + 35 x^{3} \cdot 1^{4} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.5

Multiply $35$ by $1$ .

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} \cdot 1^{4} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.6

One to any power is one.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} \cdot 1 + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.7

Multiply $35$ by $1$ .

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} \cdot 1^{5} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.8

One to any power is one.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} \cdot 1 + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.9

Multiply $21$ by $1$ .

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x \cdot 1^{6} + 1^{7}$

Step 3.4.2.1.2.10

One to any power is one.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x \cdot 1 + 1^{7}$

Step 3.4.2.1.2.11

Multiply $7$ by $1$ .

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1^{7}$

Step 3.4.2.1.2.12

One to any power is one.

$\frac{y^{5}}{7} = x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Multiply both sides of the equation by $7$ .

$7 \frac{y^{5}}{7} = 7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1)$

Step 3.5.2

Simplify both sides of the equation.

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Step 3.5.2.1

Simplify the left side.

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Step 3.5.2.1.1

Cancel the common factor of $7$ .

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Step 3.5.2.1.1.1

Cancel the common factor.

$7 \frac{y^{5}}{7} = 7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1)$

Step 3.5.2.1.1.2

Rewrite the expression.

$y^{5} = 7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1)$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Simplify $7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1)$ .

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Step 3.5.2.2.1.1

Apply the distributive property.

$y^{5} = 7 x^{7} + 7 (7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 \cdot 1$

Step 3.5.2.2.1.2

Simplify.

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Step 3.5.2.2.1.2.1

Multiply $7$ by $7$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 \cdot 1$

Step 3.5.2.2.1.2.2

Multiply $21$ by $7$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 147 x^{5} + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 \cdot 1$

Step 3.5.2.2.1.2.3

Multiply $35$ by $7$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 \cdot 1$

Step 3.5.2.2.1.2.4

Multiply $35$ by $7$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 7 (21 x^{2}) + 7 (7 x) + 7 \cdot 1$

Step 3.5.2.2.1.2.5

Multiply $21$ by $7$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 7 (7 x) + 7 \cdot 1$

Step 3.5.2.2.1.2.6

Multiply $7$ by $7$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7 \cdot 1$

Step 3.5.2.2.1.2.7

Multiply $7$ by $1$ .

$y^{5} = 7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7$

Step 3.5.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[5]{7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7}$

Step 3.5.4

Simplify $\sqrt[5]{7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7}$ .

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Step 3.5.4.1

Factor $7$ out of $7 x^{7} + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7$ .

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Step 3.5.4.1.1

Factor $7$ out of $7 x^{7}$ .

$y = \sqrt[5]{7 (x^{7}) + 49 x^{6} + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7}$

Step 3.5.4.1.2

Factor $7$ out of $49 x^{6}$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 147 x^{5} + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7}$

Step 3.5.4.1.3

Factor $7$ out of $147 x^{5}$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 7 (21 x^{5}) + 245 x^{4} + 245 x^{3} + 147 x^{2} + 49 x + 7}$

Step 3.5.4.1.4

Factor $7$ out of $245 x^{4}$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 245 x^{3} + 147 x^{2} + 49 x + 7}$

Step 3.5.4.1.5

Factor $7$ out of $245 x^{3}$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 147 x^{2} + 49 x + 7}$

Step 3.5.4.1.6

Factor $7$ out of $147 x^{2}$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 49 x + 7}$

Step 3.5.4.1.7

Factor $7$ out of $49 x$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7}$

Step 3.5.4.1.8

Factor $7$ out of $7$ .

$y = \sqrt[5]{7 (x^{7}) + 7 (7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 (1)}$

Step 3.5.4.1.9

Factor $7$ out of $7 (x^{7}) + 7 (7 x^{6})$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6}) + 7 (21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 (1)}$

Step 3.5.4.1.10

Factor $7$ out of $7 (x^{7} + 7 x^{6}) + 7 (21 x^{5})$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6} + 21 x^{5}) + 7 (35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 (1)}$

Step 3.5.4.1.11

Factor $7$ out of $7 (x^{7} + 7 x^{6} + 21 x^{5}) + 7 (35 x^{4})$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4}) + 7 (35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 (1)}$

Step 3.5.4.1.12

Factor $7$ out of $7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4}) + 7 (35 x^{3})$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3}) + 7 (21 x^{2}) + 7 (7 x) + 7 (1)}$

Step 3.5.4.1.13

Factor $7$ out of $7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3}) + 7 (21 x^{2})$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2}) + 7 (7 x) + 7 (1)}$

Step 3.5.4.1.14

Factor $7$ out of $7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2}) + 7 (7 x)$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x) + 7 (1)}$

Step 3.5.4.1.15

Factor $7$ out of $7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x) + 7 (1)$ .

$y = \sqrt[5]{7 (x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1)}$

Step 3.5.4.2

Factor $x^{7} + 7 x^{6} + 21 x^{5} + 35 x^{4} + 35 x^{3} + 21 x^{2} + 7 x + 1$ using the binomial theorem.

$y = \sqrt[5]{7 {(x + 1)}^{7}}$

Step 3.5.4.3

Rewrite $7 {(x + 1)}^{7}$ as ${(x + 1^{5})}^{5} \cdot (7 {(x + 1)}^{2})$ .

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Step 3.5.4.3.1

Factor out ${(x + 1)}^{5}$ .

$y = \sqrt[5]{7 ({(x + 1)}^{5} {(x + 1)}^{2})}$

Step 3.5.4.3.2

Reorder $7$ and ${(x + 1)}^{5}$ .

$y = \sqrt[5]{{(x + 1)}^{5} \cdot 7 {(x + 1)}^{2}}$

Step 3.5.4.3.3

Rewrite $1$ as $1^{5}$ .

$y = \sqrt[5]{{(x + 1^{5})}^{5} \cdot 7 {(x + 1)}^{2}}$

Step 3.5.4.3.4

Add parentheses.

$y = \sqrt[5]{{(x + 1^{5})}^{5} \cdot (7 {(x + 1)}^{2})}$

Step 3.5.4.4

Pull terms out from under the radical.

$y = (x + 1^{5}) \sqrt[5]{7 {(x + 1)}^{2}}$

Step 3.5.4.5

One to any power is one.

$y = (x + 1) \sqrt[5]{7 {(x + 1)}^{2}}$

Step 3.5.5

Simplify $(x + 1) \sqrt[5]{7 {(x + 1)}^{2}}$ .

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Step 3.5.5.1

Apply the distributive property.

$y = x \sqrt[5]{7 {(x + 1)}^{2}} + 1 \sqrt[5]{7 {(x + 1)}^{2}}$

Step 3.5.5.2

Multiply $\sqrt[5]{7 {(x + 1)}^{2}}$ by $1$ .

$y = x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}$

Step 4

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

$f^{- 1} (x) = x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}$

Step 5

Verify if $f^{- 1} (x) = x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}$ is the inverse of $f (x) = {(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) \sqrt[5]{7 {(({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) + 1)}^{2}} + \sqrt[5]{7 {(({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) + 1)}^{2}}$

Step 5.2.3

Combine the opposite terms in $({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) \sqrt[5]{7 {(({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) + 1)}^{2}} + \sqrt[5]{7 {(({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) + 1)}^{2}}$ .

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Step 5.2.3.1

Add $- 1$ and $1$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}} + 0)}^{2}} + \sqrt[5]{7 {(({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) + 1)}^{2}}$

Step 5.2.3.2

Add ${(\frac{x^{5}}{7})}^{\frac{1}{7}}$ and $0$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}} + \sqrt[5]{7 {(({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) + 1)}^{2}}$

Step 5.2.3.3

Add $- 1$ and $1$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}} + 0)}^{2}}$

Step 5.2.3.4

Add ${(\frac{x^{5}}{7})}^{\frac{1}{7}}$ and $0$ .

Step 5.2.4

Simplify each term.

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Step 5.2.4.1

Simplify each term.

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Step 5.2.4.1.1

Apply the product rule to $\frac{x^{5}}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{{(x^{5})}^{\frac{1}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.1.2

Multiply the exponents in ${(x^{5})}^{\frac{1}{7}}$ .

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Step 5.2.4.1.2.1

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{5 (\frac{1}{7})}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.1.2.2

Combine $5$ and $\frac{1}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.2

Multiply the exponents in ${({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}$ .

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Step 5.2.4.2.1

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 {(\frac{x^{5}}{7})}^{\frac{1}{7} \cdot 2}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.2.2

Combine $\frac{1}{7}$ and $2$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 {(\frac{x^{5}}{7})}^{\frac{2}{7}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.3

Apply the product rule to $\frac{x^{5}}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 (\frac{{(x^{5})}^{\frac{2}{7}}}{7^{\frac{2}{7}}})} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.4

Multiply the exponents in ${(x^{5})}^{\frac{2}{7}}$ .

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Step 5.2.4.4.1

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 (\frac{x^{5 (\frac{2}{7})}}{7^{\frac{2}{7}}})} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.4.2

Multiply $5 (\frac{2}{7})$ .

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Step 5.2.4.4.2.1

Combine $5$ and $\frac{2}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 (\frac{x^{\frac{5 \cdot 2}{7}}}{7^{\frac{2}{7}}})} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.4.2.2

Multiply $5$ by $2$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 (\frac{x^{\frac{10}{7}}}{7^{\frac{2}{7}}})} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.5

Combine $7$ and $\frac{x^{\frac{10}{7}}}{7^{\frac{2}{7}}}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{\frac{7 x^{\frac{10}{7}}}{7^{\frac{2}{7}}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.6

Reduce the expression by cancelling the common factors.

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Step 5.2.4.6.1

Move $7^{\frac{2}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7 x^{\frac{10}{7}} \cdot 7^{- \frac{2}{7}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.6.2

Multiply $7$ by $7^{- \frac{2}{7}}$ by adding the exponents.

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Step 5.2.4.6.2.1

Move $7^{- \frac{2}{7}}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7^{- \frac{2}{7}} \cdot (7 x^{\frac{10}{7}})} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.6.2.2

Multiply $7^{- \frac{2}{7}}$ by $7$ .

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Step 5.2.4.6.2.2.1

Raise $7$ to the power of $1$ .

Step 5.2.4.6.2.2.2

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7^{- \frac{2}{7} + 1} x^{\frac{10}{7}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.6.2.3

Write $1$ as a fraction with a common denominator.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7^{- \frac{2}{7} + \frac{7}{7}} x^{\frac{10}{7}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.6.2.4

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7^{\frac{- 2 + 7}{7}} x^{\frac{10}{7}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.6.2.5

Add $- 2$ and $7$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{7^{\frac{5}{7}} x^{\frac{10}{7}}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.7

Rewrite $7^{\frac{5}{7}} x^{\frac{10}{7}}$ as ${(7^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) \sqrt[5]{{(7^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.8

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} - 1) (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.9

Apply the distributive property.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = \frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}} \cdot (7^{\frac{1}{7}} x^{\frac{2}{7}}) - 1 (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.10

Rewrite using the commutative property of multiplication.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = 7^{\frac{1}{7}} (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}}) \cdot x^{\frac{2}{7}} - 1 (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.11

Rewrite $- 1 (7^{\frac{1}{7}} x^{\frac{2}{7}})$ as $- (7^{\frac{1}{7}} x^{\frac{2}{7}})$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = 7^{\frac{1}{7}} (\frac{x^{\frac{5}{7}}}{7^{\frac{1}{7}}}) \cdot x^{\frac{2}{7}} - (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.12

Simplify each term.

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Step 5.2.4.12.1

Cancel the common factor of $7^{\frac{1}{7}}$ .

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Step 5.2.4.12.1.1

Cancel the common factor.

Step 5.2.4.12.1.2

Rewrite the expression.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x^{\frac{5}{7}} x^{\frac{2}{7}} - (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.12.2

Multiply $x^{\frac{5}{7}}$ by $x^{\frac{2}{7}}$ by adding the exponents.

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Step 5.2.4.12.2.1

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x^{\frac{5}{7} + \frac{2}{7}} - (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.12.2.2

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x^{\frac{5 + 2}{7}} - (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.12.2.3

Add $5$ and $2$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x^{\frac{7}{7}} - (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.12.2.4

Divide $7$ by $7$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - (7^{\frac{1}{7}} x^{\frac{2}{7}}) + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.12.3

Simplify $x^{1}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 {({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}}$

Step 5.2.4.13

Multiply the exponents in ${({(\frac{x^{5}}{7})}^{\frac{1}{7}})}^{2}$ .

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Step 5.2.4.13.1

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 {(\frac{x^{5}}{7})}^{\frac{1}{7} \cdot 2}}$

Step 5.2.4.13.2

Combine $\frac{1}{7}$ and $2$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 {(\frac{x^{5}}{7})}^{\frac{2}{7}}}$

Step 5.2.4.14

Apply the product rule to $\frac{x^{5}}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 (\frac{{(x^{5})}^{\frac{2}{7}}}{7^{\frac{2}{7}}})}$

Step 5.2.4.15

Multiply the exponents in ${(x^{5})}^{\frac{2}{7}}$ .

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Step 5.2.4.15.1

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 (\frac{x^{5 (\frac{2}{7})}}{7^{\frac{2}{7}}})}$

Step 5.2.4.15.2

Multiply $5 (\frac{2}{7})$ .

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Step 5.2.4.15.2.1

Combine $5$ and $\frac{2}{7}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 (\frac{x^{\frac{5 \cdot 2}{7}}}{7^{\frac{2}{7}}})}$

Step 5.2.4.15.2.2

Multiply $5$ by $2$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 (\frac{x^{\frac{10}{7}}}{7^{\frac{2}{7}}})}$

Step 5.2.4.16

Combine $7$ and $\frac{x^{\frac{10}{7}}}{7^{\frac{2}{7}}}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{\frac{7 x^{\frac{10}{7}}}{7^{\frac{2}{7}}}}$

Step 5.2.4.17

Reduce the expression by cancelling the common factors.

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Step 5.2.4.17.1

Move $7^{\frac{2}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7 x^{\frac{10}{7}} \cdot 7^{- \frac{2}{7}}}$

Step 5.2.4.17.2

Multiply $7$ by $7^{- \frac{2}{7}}$ by adding the exponents.

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Step 5.2.4.17.2.1

Move $7^{- \frac{2}{7}}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7^{- \frac{2}{7}} \cdot (7 x^{\frac{10}{7}})}$

Step 5.2.4.17.2.2

Multiply $7^{- \frac{2}{7}}$ by $7$ .

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Step 5.2.4.17.2.2.1

Raise $7$ to the power of $1$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7^{- \frac{2}{7}} \cdot (7 x^{\frac{10}{7}})}$

Step 5.2.4.17.2.2.2

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7^{- \frac{2}{7} + 1} x^{\frac{10}{7}}}$

Step 5.2.4.17.2.3

Write $1$ as a fraction with a common denominator.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7^{- \frac{2}{7} + \frac{7}{7}} x^{\frac{10}{7}}}$

Step 5.2.4.17.2.4

Combine the numerators over the common denominator.

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7^{\frac{- 2 + 7}{7}} x^{\frac{10}{7}}}$

Step 5.2.4.17.2.5

Add $- 2$ and $7$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{7^{\frac{5}{7}} x^{\frac{10}{7}}}$

Step 5.2.4.18

Rewrite $7^{\frac{5}{7}} x^{\frac{10}{7}}$ as ${(7^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}$ .

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + \sqrt[5]{{(7^{\frac{1}{7}} x^{\frac{2}{7}})}^{5}}$

Step 5.2.4.19

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

$f^{- 1} ({(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1) = x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + 7^{\frac{1}{7}} x^{\frac{2}{7}}$

Step 5.2.5

Combine the opposite terms in $x - 7^{\frac{1}{7}} x^{\frac{2}{7}} + 7^{\frac{1}{7}} x^{\frac{2}{7}}$ .

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Step 5.2.5.1

Add $- 7^{\frac{1}{7}} x^{\frac{2}{7}}$ and $7^{\frac{1}{7}} x^{\frac{2}{7}}$ .

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

Step 5.2.5.2

Add $x$ and $0$ .

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

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}})}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

Factor $\sqrt[5]{7 {(x + 1)}^{2}}$ out of $x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}$ .

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Step 5.3.3.1.1.1

Factor $\sqrt[5]{7 {(x + 1)}^{2}}$ out of $x \sqrt[5]{7 {(x + 1)}^{2}}$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(\sqrt[5]{7 {(x + 1)}^{2}} x + \sqrt[5]{7 {(x + 1)}^{2}})}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.1.2

Multiply by $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(\sqrt[5]{7 {(x + 1)}^{2}} x + \sqrt[5]{7 {(x + 1)}^{2}} \cdot 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.1.3

Factor $\sqrt[5]{7 {(x + 1)}^{2}}$ out of $\sqrt[5]{7 {(x + 1)}^{2}} x + \sqrt[5]{7 {(x + 1)}^{2}} \cdot 1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(\sqrt[5]{7 {(x + 1)}^{2}} (x + 1))}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.2

Apply the product rule to $\sqrt[5]{7 {(x + 1)}^{2}} (x + 1)$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{\sqrt[5]{7 {(x + 1)}^{2}}}^{5} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.3

Rewrite ${\sqrt[5]{7 {(x + 1)}^{2}}}^{5}$ as $7 {(x + 1)}^{2}$ .

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Step 5.3.3.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{7 {(x + 1)}^{2}}$ as ${(7 {(x + 1)}^{2})}^{\frac{1}{5}}$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{({(7 {(x + 1)}^{2})}^{\frac{1}{5}})}^{5} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.3.2

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

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(7 {(x + 1)}^{2})}^{\frac{1}{5} \cdot 5} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.3.3

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

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(7 {(x + 1)}^{2})}^{\frac{5}{5}} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.3.4

Cancel the common factor of $5$ .

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Step 5.3.3.1.3.4.1

Cancel the common factor.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{{(7 {(x + 1)}^{2})}^{\frac{5}{5}} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.3.4.2

Rewrite the expression.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 {(x + 1)}^{2}) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.3.5

Simplify.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 {(x + 1)}^{2} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.4

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 ((x + 1) (x + 1)) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.5

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 5.3.3.1.5.1

Apply the distributive property.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x (x + 1) + 1 (x + 1)) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.5.2

Apply the distributive property.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x \cdot x + x \cdot 1 + 1 (x + 1)) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.5.3

Apply the distributive property.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.6

Simplify and combine like terms.

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Step 5.3.3.1.6.1

Simplify each term.

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Step 5.3.3.1.6.1.1

Multiply $x$ by $x$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + x \cdot 1 + 1 x + 1 \cdot 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.6.1.2

Multiply $x$ by $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + x + 1 x + 1 \cdot 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.6.1.3

Multiply $x$ by $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + x + x + 1 \cdot 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.6.1.4

Multiply $1$ by $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + x + x + 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.6.2

Add $x$ and $x$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + 2 x + 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.7

Apply the distributive property.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 x^{2} + 7 (2 x) + 7 \cdot 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.8

Simplify.

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Step 5.3.3.1.8.1

Multiply $2$ by $7$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 x^{2} + 14 x + 7 \cdot 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.8.2

Multiply $7$ by $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 x^{2} + 14 x + 7) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.9

Factor $7$ out of $7 x^{2} + 14 x + 7$ .

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Step 5.3.3.1.9.1

Factor $7$ out of $7 x^{2}$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 (x^{2}) + 14 x + 7) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.9.2

Factor $7$ out of $14 x$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 (x^{2}) + 7 (2 x) + 7) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.9.3

Factor $7$ out of $7$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 (x^{2}) + 7 (2 x) + 7 (1)) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.9.4

Factor $7$ out of $7 (x^{2}) + 7 (2 x)$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{(7 (x^{2} + 2 x) + 7 (1)) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.9.5

Factor $7$ out of $7 (x^{2} + 2 x) + 7 (1)$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + 2 x + 1) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.10

Factor using the perfect square rule.

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Step 5.3.3.1.10.1

Rewrite $1$ as $1^{2}$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + 2 x + 1^{2}) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.10.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 5.3.3.1.10.3

Rewrite the polynomial.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 (x^{2} + 2 \cdot x \cdot 1 + 1^{2}) {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.10.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 {(x + 1)}^{2} {(x + 1)}^{5}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.11

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

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Step 5.3.3.1.11.1

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

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 ({(x + 1)}^{5} {(x + 1)}^{2})}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.11.2

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

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 {(x + 1)}^{5 + 2}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.1.11.3

Add $5$ and $2$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 {(x + 1)}^{7}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.2

Cancel the common factor of $7$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(\frac{7 {(x + 1)}^{7}}{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.2.2

Divide ${(x + 1)}^{7}$ by $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {({(x + 1)}^{7})}^{\frac{1}{7}} - 1$

Step 5.3.3.3

Multiply the exponents in ${({(x + 1)}^{7})}^{\frac{1}{7}}$ .

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Step 5.3.3.3.1

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

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(x + 1)}^{7 (\frac{1}{7})} - 1$

Step 5.3.3.3.2

Cancel the common factor of $7$ .

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Step 5.3.3.3.2.1

Cancel the common factor.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = {(x + 1)}^{7 (\frac{1}{7})} - 1$

Step 5.3.3.3.2.2

Rewrite the expression.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = (x + 1) - 1$

Step 5.3.3.4

Simplify.

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = x + 1 - 1$

Step 5.3.4

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

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Step 5.3.4.1

Subtract $1$ from $1$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}$ is the inverse of $f (x) = {(\frac{x^{5}}{7})}^{\frac{1}{7}} - 1$ .

$f^{- 1} (x) = x \sqrt[5]{7 {(x + 1)}^{2}} + \sqrt[5]{7 {(x + 1)}^{2}}$