Precalculus Examples

$f (x) = \sqrt[3]{\frac{x}{8}} - 4$

Step 1

Write $f (x) = \sqrt[3]{\frac{x}{8}} - 4$ as an equation.

$y = \sqrt[3]{\frac{x}{8}} - 4$

Step 2

Interchange the variables.

$x = \sqrt[3]{\frac{y}{8}} - 4$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\sqrt[3]{\frac{y}{8}} - 4 = x$ .

$\sqrt[3]{\frac{y}{8}} - 4 = x$

Step 3.2

Add $4$ to both sides of the equation.

$\sqrt[3]{\frac{y}{8}} = x + 4$

Step 3.3

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{\frac{y}{8}}}^{3} = {(x + 4)}^{3}$

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 $\sqrt[3]{\frac{y}{8}}$ as ${(\frac{y}{8})}^{\frac{1}{3}}$ .

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

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Simplify ${({(\frac{y}{8})}^{\frac{1}{3}})}^{3}$ .

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Step 3.4.2.1.1

Multiply the exponents in ${({(\frac{y}{8})}^{\frac{1}{3}})}^{3}$ .

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Step 3.4.2.1.1.1

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

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

Step 3.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

$\frac{y}{8} = {(x + 4)}^{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(x + 4)}^{3}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$\frac{y}{8} = x^{3} + 3 x^{2} \cdot 4 + 3 x \cdot 4^{2} + 4^{3}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Multiply $4$ by $3$ .

$\frac{y}{8} = x^{3} + 12 x^{2} + 3 x \cdot 4^{2} + 4^{3}$

Step 3.4.3.1.2.2

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

$\frac{y}{8} = x^{3} + 12 x^{2} + 3 x \cdot 16 + 4^{3}$

Step 3.4.3.1.2.3

Multiply $16$ by $3$ .

$\frac{y}{8} = x^{3} + 12 x^{2} + 48 x + 4^{3}$

Step 3.4.3.1.2.4

Raise $4$ to the power of $3$ .

$\frac{y}{8} = x^{3} + 12 x^{2} + 48 x + 64$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Multiply both sides of the equation by $8$ .

$8 \frac{y}{8} = 8 (x^{3} + 12 x^{2} + 48 x + 64)$

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 $8$ .

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Step 3.5.2.1.1.1

Cancel the common factor.

$8 \frac{y}{8} = 8 (x^{3} + 12 x^{2} + 48 x + 64)$

Step 3.5.2.1.1.2

Rewrite the expression.

$y = 8 (x^{3} + 12 x^{2} + 48 x + 64)$

Step 3.5.2.2

Simplify the right side.

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Step 3.5.2.2.1

Simplify $8 (x^{3} + 12 x^{2} + 48 x + 64)$ .

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Step 3.5.2.2.1.1

Apply the distributive property.

$y = 8 x^{3} + 8 (12 x^{2}) + 8 (48 x) + 8 \cdot 64$

Step 3.5.2.2.1.2

Simplify.

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Step 3.5.2.2.1.2.1

Multiply $12$ by $8$ .

$y = 8 x^{3} + 96 x^{2} + 8 (48 x) + 8 \cdot 64$

Step 3.5.2.2.1.2.2

Multiply $48$ by $8$ .

$y = 8 x^{3} + 96 x^{2} + 384 x + 8 \cdot 64$

Step 3.5.2.2.1.2.3

Multiply $8$ by $64$ .

$y = 8 x^{3} + 96 x^{2} + 384 x + 512$

Step 4

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

$f^{- 1} (x) = 8 x^{3} + 96 x^{2} + 384 x + 512$

Step 5

Verify if $f^{- 1} (x) = 8 x^{3} + 96 x^{2} + 384 x + 512$ is the inverse of $f (x) = \sqrt[3]{\frac{x}{8}} - 4$ .

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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 $f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 {(\sqrt[3]{\frac{x}{8}} - 4)}^{3} + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Rewrite $\frac{x}{8}$ as ${(\frac{1}{2})}^{3} x$ .

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Step 5.2.3.1.1.1

Factor the perfect power $1^{3}$ out of $x$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 {(\sqrt[3]{\frac{1^{3} x}{8}} - 4)}^{3} + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.1.1.2

Factor the perfect power $2^{3}$ out of $8$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 {(\sqrt[3]{\frac{1^{3} x}{2^{3} \cdot 1}} - 4)}^{3} + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.1.1.3

Rearrange the fraction $\frac{1^{3} x}{2^{3} \cdot 1}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 {(\sqrt[3]{{(\frac{1}{2})}^{3} x} - 4)}^{3} + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.1.2

Pull terms out from under the radical.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 {(\frac{1}{2} \cdot \sqrt[3]{x} - 4)}^{3} + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.1.3

Combine $\frac{1}{2}$ and $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 {(\frac{\sqrt[3]{x}}{2} - 4)}^{3} + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.2

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 ({(\frac{\sqrt[3]{x}}{2})}^{3} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3

Simplify each term.

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Step 5.2.3.3.1

Apply the product rule to $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{{\sqrt[3]{x}}^{3}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.2

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 5.2.3.3.2.1

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{{(x^{\frac{1}{3}})}^{3}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.2.2

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x^{\frac{1}{3} \cdot 3}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.2.3

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x^{\frac{3}{3}}}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.2.4

Cancel the common factor of $3$ .

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Step 5.2.3.3.2.4.1

Cancel the common factor.

Step 5.2.3.3.2.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{2^{3}} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.2.5

Simplify.

Step 5.2.3.3.3

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + 3 {(\frac{\sqrt[3]{x}}{2})}^{2} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.4

Apply the product rule to $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + 3 (\frac{{\sqrt[3]{x}}^{2}}{2^{2}}) \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.5

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + 3 (\frac{\sqrt[3]{x^{2}}}{2^{2}}) \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.6

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + 3 (\frac{\sqrt[3]{x^{2}}}{4}) \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.7

Combine $3$ and $\frac{\sqrt[3]{x^{2}}}{4}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + \frac{3 \sqrt[3]{x^{2}}}{4} \cdot - 4 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.8

Cancel the common factor of $4$ .

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Step 5.2.3.3.8.1

Factor $4$ out of $- 4$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + \frac{3 \sqrt[3]{x^{2}}}{4} \cdot (4 (- 1)) + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.8.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + \frac{3 \sqrt[3]{x^{2}}}{4} \cdot (4 \cdot - 1) + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.8.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} + 3 \sqrt[3]{x^{2}} \cdot - 1 + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.9

Multiply $- 1$ by $3$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + 3 (\frac{\sqrt[3]{x}}{2}) \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.10

Combine $3$ and $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + \frac{3 \sqrt[3]{x}}{2} \cdot {(- 4)}^{2} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.11

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + \frac{3 \sqrt[3]{x}}{2} \cdot 16 + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.12

Cancel the common factor of $2$ .

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Step 5.2.3.3.12.1

Factor $2$ out of $16$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + \frac{3 \sqrt[3]{x}}{2} \cdot (2 (8)) + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.12.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + \frac{3 \sqrt[3]{x}}{2} \cdot (2 \cdot 8) + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.12.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 8 + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.13

Multiply $8$ by $3$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + 24 \sqrt[3]{x} + {(- 4)}^{3}) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.3.14

Raise $- 4$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8} - 3 \sqrt[3]{x^{2}} + 24 \sqrt[3]{x} - 64) + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.4

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8}) + 8 (- 3 \sqrt[3]{x^{2}}) + 8 (24 \sqrt[3]{x}) + 8 \cdot - 64 + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.5

Simplify.

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Step 5.2.3.5.1

Cancel the common factor of $8$ .

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Step 5.2.3.5.1.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = 8 (\frac{x}{8}) + 8 (- 3 \sqrt[3]{x^{2}}) + 8 (24 \sqrt[3]{x}) + 8 \cdot - 64 + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.5.1.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 8 (- 3 \sqrt[3]{x^{2}}) + 8 (24 \sqrt[3]{x}) + 8 \cdot - 64 + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.5.2

Multiply $- 3$ by $8$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 8 (24 \sqrt[3]{x}) + 8 \cdot - 64 + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.5.3

Multiply $24$ by $8$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} + 8 \cdot - 64 + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.5.4

Multiply $8$ by $- 64$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 {(\sqrt[3]{\frac{x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.6

Simplify each term.

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Step 5.2.3.6.1

Rewrite $\frac{x}{8}$ as ${(\frac{1}{2})}^{3} x$ .

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Step 5.2.3.6.1.1

Factor the perfect power $1^{3}$ out of $x$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 {(\sqrt[3]{\frac{1^{3} x}{8}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.6.1.2

Factor the perfect power $2^{3}$ out of $8$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 {(\sqrt[3]{\frac{1^{3} x}{2^{3} \cdot 1}} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.6.1.3

Rearrange the fraction $\frac{1^{3} x}{2^{3} \cdot 1}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 {(\sqrt[3]{{(\frac{1}{2})}^{3} x} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.6.2

Pull terms out from under the radical.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 {(\frac{1}{2} \cdot \sqrt[3]{x} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.6.3

Combine $\frac{1}{2}$ and $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 {(\frac{\sqrt[3]{x}}{2} - 4)}^{2} + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.7

Rewrite ${(\frac{\sqrt[3]{x}}{2} - 4)}^{2}$ as $(\frac{\sqrt[3]{x}}{2} - 4) (\frac{\sqrt[3]{x}}{2} - 4)$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 ((\frac{\sqrt[3]{x}}{2} - 4) (\frac{\sqrt[3]{x}}{2} - 4)) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.8

Expand $(\frac{\sqrt[3]{x}}{2} - 4) (\frac{\sqrt[3]{x}}{2} - 4)$ using the FOIL Method.

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Step 5.2.3.8.1

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x}}{2} \cdot (\frac{\sqrt[3]{x}}{2} - 4) - 4 (\frac{\sqrt[3]{x}}{2} - 4)) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.8.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x}}{2} \cdot \frac{\sqrt[3]{x}}{2} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 (\frac{\sqrt[3]{x}}{2} - 4)) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.8.3

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x}}{2} \cdot \frac{\sqrt[3]{x}}{2} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9

Simplify and combine like terms.

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Step 5.2.3.9.1

Simplify each term.

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Step 5.2.3.9.1.1

Multiply $\frac{\sqrt[3]{x}}{2} \cdot \frac{\sqrt[3]{x}}{2}$ .

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Step 5.2.3.9.1.1.1

Multiply $\frac{\sqrt[3]{x}}{2}$ by $\frac{\sqrt[3]{x}}{2}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x} \sqrt[3]{x}}{2 \cdot 2} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.1.2

Raise $\sqrt[3]{x}$ to the power of $1$ .

Step 5.2.3.9.1.1.3

Raise $\sqrt[3]{x}$ to the power of $1$ .

Step 5.2.3.9.1.1.4

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

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{{\sqrt[3]{x}}^{1 + 1}}{2 \cdot 2} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.1.5

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{{\sqrt[3]{x}}^{2}}{2 \cdot 2} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.1.6

Multiply $2$ by $2$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{{\sqrt[3]{x}}^{2}}{4} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.2

Rewrite ${\sqrt[3]{x}}^{2}$ as $\sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{\sqrt[3]{x}}{2} \cdot - 4 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.3

Cancel the common factor of $2$ .

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Step 5.2.3.9.1.3.1

Factor $2$ out of $- 4$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{\sqrt[3]{x}}{2} \cdot (2 (- 2)) - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.3.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} + \frac{\sqrt[3]{x}}{2} \cdot (2 \cdot - 2) - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.3.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} + \sqrt[3]{x} \cdot - 2 - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.4

Move $- 2$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} - 2 \cdot \sqrt[3]{x} - 4 \frac{\sqrt[3]{x}}{2} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.5

Cancel the common factor of $2$ .

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Step 5.2.3.9.1.5.1

Factor $2$ out of $- 4$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} - 2 \sqrt[3]{x} + 2 (- 2) (\frac{\sqrt[3]{x}}{2}) - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.5.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} - 2 \sqrt[3]{x} + 2 \cdot (- 2 \frac{\sqrt[3]{x}}{2}) - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.5.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} - 2 \sqrt[3]{x} - 2 \sqrt[3]{x} - 4 \cdot - 4) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.1.6

Multiply $- 4$ by $- 4$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} - 2 \sqrt[3]{x} - 2 \sqrt[3]{x} + 16) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.9.2

Subtract $2 \sqrt[3]{x}$ from $- 2 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4} - 4 \sqrt[3]{x} + 16) + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.10

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 96 (\frac{\sqrt[3]{x^{2}}}{4}) + 96 (- 4 \sqrt[3]{x}) + 96 \cdot 16 + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.11

Simplify.

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Step 5.2.3.11.1

Cancel the common factor of $4$ .

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Step 5.2.3.11.1.1

Factor $4$ out of $96$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 4 (24) (\frac{\sqrt[3]{x^{2}}}{4}) + 96 (- 4 \sqrt[3]{x}) + 96 \cdot 16 + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.11.1.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 4 \cdot (24 (\frac{\sqrt[3]{x^{2}}}{4})) + 96 (- 4 \sqrt[3]{x}) + 96 \cdot 16 + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.11.1.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} + 96 (- 4 \sqrt[3]{x}) + 96 \cdot 16 + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.11.2

Multiply $- 4$ by $96$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 96 \cdot 16 + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.11.3

Multiply $96$ by $16$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\sqrt[3]{\frac{x}{8}} - 4) + 512$

Step 5.2.3.12

Simplify each term.

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Step 5.2.3.12.1

Rewrite $\frac{x}{8}$ as ${(\frac{1}{2})}^{3} x$ .

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Step 5.2.3.12.1.1

Factor the perfect power $1^{3}$ out of $x$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\sqrt[3]{\frac{1^{3} x}{8}} - 4) + 512$

Step 5.2.3.12.1.2

Factor the perfect power $2^{3}$ out of $8$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\sqrt[3]{\frac{1^{3} x}{2^{3} \cdot 1}} - 4) + 512$

Step 5.2.3.12.1.3

Rearrange the fraction $\frac{1^{3} x}{2^{3} \cdot 1}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\sqrt[3]{{(\frac{1}{2})}^{3} x} - 4) + 512$

Step 5.2.3.12.2

Pull terms out from under the radical.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\frac{1}{2} \cdot \sqrt[3]{x} - 4) + 512$

Step 5.2.3.12.3

Combine $\frac{1}{2}$ and $\sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\frac{\sqrt[3]{x}}{2} - 4) + 512$

Step 5.2.3.13

Apply the distributive property.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 384 (\frac{\sqrt[3]{x}}{2}) + 384 \cdot - 4 + 512$

Step 5.2.3.14

Cancel the common factor of $2$ .

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Step 5.2.3.14.1

Factor $2$ out of $384$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 2 (192) (\frac{\sqrt[3]{x}}{2}) + 384 \cdot - 4 + 512$

Step 5.2.3.14.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 2 \cdot (192 (\frac{\sqrt[3]{x}}{2})) + 384 \cdot - 4 + 512$

Step 5.2.3.14.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 192 \sqrt[3]{x} + 384 \cdot - 4 + 512$

Step 5.2.3.15

Multiply $384$ by $- 4$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 192 \sqrt[3]{x} - 1536 + 512$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x - 24 \sqrt[3]{x^{2}} + 192 \sqrt[3]{x} - 512 + 24 \sqrt[3]{x^{2}} - 384 \sqrt[3]{x} + 1536 + 192 \sqrt[3]{x} - 1536 + 512$ .

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Step 5.2.4.1.1

Add $- 24 \sqrt[3]{x^{2}}$ and $24 \sqrt[3]{x^{2}}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 0 + 192 \sqrt[3]{x} - 512 - 384 \sqrt[3]{x} + 1536 + 192 \sqrt[3]{x} - 1536 + 512$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 192 \sqrt[3]{x} - 512 - 384 \sqrt[3]{x} + 1536 + 192 \sqrt[3]{x} - 1536 + 512$

Step 5.2.4.1.3

Subtract $1536$ from $1536$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 192 \sqrt[3]{x} - 512 - 384 \sqrt[3]{x} + 0 + 192 \sqrt[3]{x} + 512$

Step 5.2.4.1.4

Add $x + 192 \sqrt[3]{x} - 512 - 384 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 192 \sqrt[3]{x} - 512 - 384 \sqrt[3]{x} + 192 \sqrt[3]{x} + 512$

Step 5.2.4.1.5

Add $- 512$ and $512$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 192 \sqrt[3]{x} + 0 - 384 \sqrt[3]{x} + 192 \sqrt[3]{x}$

Step 5.2.4.1.6

Add $x + 192 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 192 \sqrt[3]{x} - 384 \sqrt[3]{x} + 192 \sqrt[3]{x}$

Step 5.2.4.2

Subtract $384 \sqrt[3]{x}$ from $192 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x - 192 \sqrt[3]{x} + 192 \sqrt[3]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 192 \sqrt[3]{x} + 192 \sqrt[3]{x}$ .

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Step 5.2.4.3.1

Add $- 192 \sqrt[3]{x}$ and $192 \sqrt[3]{x}$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (\sqrt[3]{\frac{x}{8}} - 4) = x$

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 $f (8 x^{3} + 96 x^{2} + 384 x + 512)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 x^{3} + 96 x^{2} + 384 x + 512}{8}} - 4$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Factor $8$ out of $8 x^{3} + 96 x^{2} + 384 x + 512$ .

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Step 5.3.3.1.1

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

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3}) + 96 x^{2} + 384 x + 512}{8}} - 4$

Step 5.3.3.1.2

Factor $8$ out of $96 x^{2}$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3}) + 8 (12 x^{2}) + 384 x + 512}{8}} - 4$

Step 5.3.3.1.3

Factor $8$ out of $384 x$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3}) + 8 (12 x^{2}) + 8 (48 x) + 512}{8}} - 4$

Step 5.3.3.1.4

Factor $8$ out of $512$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 x^{3} + 8 (12 x^{2}) + 8 (48 x) + 8 \cdot 64}{8}} - 4$

Step 5.3.3.1.5

Factor $8$ out of $8 x^{3} + 8 (12 x^{2})$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3} + 12 x^{2}) + 8 (48 x) + 8 \cdot 64}{8}} - 4$

Step 5.3.3.1.6

Factor $8$ out of $8 (x^{3} + 12 x^{2}) + 8 (48 x)$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3} + 12 x^{2} + 48 x) + 8 \cdot 64}{8}} - 4$

Step 5.3.3.1.7

Factor $8$ out of $8 (x^{3} + 12 x^{2} + 48 x) + 8 \cdot 64$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3} + 12 x^{2} + 48 x + 64)}{8}} - 4$

Step 5.3.3.2

Reduce the expression by cancelling the common factors.

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Step 5.3.3.2.1

Reduce the expression $\frac{8 (x^{3} + 12 x^{2} + 48 x + 64)}{8}$ by cancelling the common factors.

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Step 5.3.3.2.1.1

Cancel the common factor.

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{8 (x^{3} + 12 x^{2} + 48 x + 64)}{8}} - 4$

Step 5.3.3.2.1.2

Rewrite the expression.

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{\frac{x^{3} + 12 x^{2} + 48 x + 64}{1}} - 4$

Step 5.3.3.2.2

Divide $x^{3} + 12 x^{2} + 48 x + 64$ by $1$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{x^{3} + 12 x^{2} + 48 x + 64} - 4$

Step 5.3.3.3

Make each term match the terms from the binomial theorem formula.

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{x^{3} + 3 \cdot (4 x^{2}) + 3 \cdot (4^{2} x) + 4^{3}} - 4$

Step 5.3.3.4

Factor $x^{3} + 3 \cdot 4 x^{2} + 3 \cdot 4^{2} x + 4^{3}$ using the binomial theorem.

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = \sqrt[3]{{(x + 4)}^{3}} - 4$

Step 5.3.3.5

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

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = x + 4 - 4$

Step 5.3.4

Combine the opposite terms in $x + 4 - 4$ .

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Step 5.3.4.1

Subtract $4$ from $4$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (8 x^{3} + 96 x^{2} + 384 x + 512) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 8 x^{3} + 96 x^{2} + 384 x + 512$ is the inverse of $f (x) = \sqrt[3]{\frac{x}{8}} - 4$ .

$f^{- 1} (x) = 8 x^{3} + 96 x^{2} + 384 x + 512$