Calculus Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Add $8$ to both sides of the equation.

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

Step 3.3

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

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

Step 3.4

Simplify each side of the equation.

Tap for more steps...

Step 3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y + 4}$ as ${(y + 4)}^{\frac{1}{3}}$ .

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

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

Simplify ${({(y + 4)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.4.2.1.1

Multiply the exponents in ${({(y + 4)}^{\frac{1}{3}})}^{3}$ .

Tap for more steps...

Step 3.4.2.1.1.1

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

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

Step 3.4.2.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.4.2.1.1.2.1

Cancel the common factor.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

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

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

Simplify ${(x + 8)}^{3}$ .

Tap for more steps...

Step 3.4.3.1.1

Use the Binomial Theorem.

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

Step 3.4.3.1.2

Simplify each term.

Tap for more steps...

Step 3.4.3.1.2.1

Multiply $8$ by $3$ .

$y + 4 = x^{3} + 24 x^{2} + 3 x \cdot 8^{2} + 8^{3}$

Step 3.4.3.1.2.2

Raise $8$ to the power of $2$ .

$y + 4 = x^{3} + 24 x^{2} + 3 x \cdot 64 + 8^{3}$

Step 3.4.3.1.2.3

Multiply $64$ by $3$ .

$y + 4 = x^{3} + 24 x^{2} + 192 x + 8^{3}$

Step 3.4.3.1.2.4

Raise $8$ to the power of $3$ .

$y + 4 = x^{3} + 24 x^{2} + 192 x + 512$

Step 3.5

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

Tap for more steps...

Step 3.5.1

Subtract $4$ from both sides of the equation.

$y = x^{3} + 24 x^{2} + 192 x + 512 - 4$

Step 3.5.2

Subtract $4$ from $512$ .

$y = x^{3} + 24 x^{2} + 192 x + 508$

Step 4

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

$f^{- 1} (x) = x^{3} + 24 x^{2} + 192 x + 508$

Step 5

Verify if $f^{- 1} (x) = x^{3} + 24 x^{2} + 192 x + 508$ is the inverse of $f (x) = \sqrt[3]{x + 4} - 8$ .

Tap for more steps...

Step 5.1

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

Step 5.2

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

Tap for more steps...

Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

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

$f^{- 1} (\sqrt[3]{x + 4} - 8) = {(\sqrt[3]{x + 4} - 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = {\sqrt[3]{x + 4}}^{3} + 3 {\sqrt[3]{x + 4}}^{2} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2

Simplify each term.

Tap for more steps...

Step 5.2.3.2.1

Rewrite ${\sqrt[3]{x + 4}}^{3}$ as $x + 4$ .

Tap for more steps...

Step 5.2.3.2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 4}$ as ${(x + 4)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = {({(x + 4)}^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x + 4}}^{2} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.1.2

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

$f^{- 1} (\sqrt[3]{x + 4} - 8) = {(x + 4)}^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x + 4}}^{2} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.1.3

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

$f^{- 1} (\sqrt[3]{x + 4} - 8) = {(x + 4)}^{\frac{3}{3}} + 3 {\sqrt[3]{x + 4}}^{2} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.1.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.2.1.4.1

Cancel the common factor.

Step 5.2.3.2.1.4.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = (x + 4) + 3 {\sqrt[3]{x + 4}}^{2} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.1.5

Simplify.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x + 4 + 3 {\sqrt[3]{x + 4}}^{2} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.2

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

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x + 4 + 3 \sqrt[3]{{(x + 4)}^{2}} \cdot - 8 + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.3

Multiply $- 8$ by $3$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x + 4 - 24 \sqrt[3]{{(x + 4)}^{2}} + 3 \sqrt[3]{x + 4} {(- 8)}^{2} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.4

Raise $- 8$ to the power of $2$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x + 4 - 24 \sqrt[3]{{(x + 4)}^{2}} + 3 \sqrt[3]{x + 4} \cdot 64 + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.5

Multiply $64$ by $3$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x + 4 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + {(- 8)}^{3} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.2.6

Raise $- 8$ to the power of $3$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x + 4 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} - 512 + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.3

Subtract $512$ from $4$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 {(\sqrt[3]{x + 4} - 8)}^{2} + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.4

Rewrite ${(\sqrt[3]{x + 4} - 8)}^{2}$ as $(\sqrt[3]{x + 4} - 8) (\sqrt[3]{x + 4} - 8)$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 ((\sqrt[3]{x + 4} - 8) (\sqrt[3]{x + 4} - 8)) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.5

Expand $(\sqrt[3]{x + 4} - 8) (\sqrt[3]{x + 4} - 8)$ using the FOIL Method.

Tap for more steps...

Step 5.2.3.5.1

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{x + 4} (\sqrt[3]{x + 4} - 8) - 8 (\sqrt[3]{x + 4} - 8)) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.5.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{x + 4} \sqrt[3]{x + 4} + \sqrt[3]{x + 4} \cdot - 8 - 8 (\sqrt[3]{x + 4} - 8)) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.5.3

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{x + 4} \sqrt[3]{x + 4} + \sqrt[3]{x + 4} \cdot - 8 - 8 \sqrt[3]{x + 4} - 8 \cdot - 8) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.6

Simplify and combine like terms.

Tap for more steps...

Step 5.2.3.6.1

Simplify each term.

Tap for more steps...

Step 5.2.3.6.1.1

Multiply $\sqrt[3]{x + 4} \sqrt[3]{x + 4}$ .

Tap for more steps...

Step 5.2.3.6.1.1.1

Raise $\sqrt[3]{x + 4}$ to the power of $1$ .

Step 5.2.3.6.1.1.2

Raise $\sqrt[3]{x + 4}$ to the power of $1$ .

Step 5.2.3.6.1.1.3

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

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 ({\sqrt[3]{x + 4}}^{1 + 1} + \sqrt[3]{x + 4} \cdot - 8 - 8 \sqrt[3]{x + 4} - 8 \cdot - 8) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.6.1.1.4

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 ({\sqrt[3]{x + 4}}^{2} + \sqrt[3]{x + 4} \cdot - 8 - 8 \sqrt[3]{x + 4} - 8 \cdot - 8) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.6.1.2

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

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{{(x + 4)}^{2}} + \sqrt[3]{x + 4} \cdot - 8 - 8 \sqrt[3]{x + 4} - 8 \cdot - 8) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.6.1.3

Move $- 8$ to the left of $\sqrt[3]{x + 4}$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{{(x + 4)}^{2}} - 8 \cdot \sqrt[3]{x + 4} - 8 \sqrt[3]{x + 4} - 8 \cdot - 8) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.6.1.4

Multiply $- 8$ by $- 8$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{{(x + 4)}^{2}} - 8 \sqrt[3]{x + 4} - 8 \sqrt[3]{x + 4} + 64) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.6.2

Subtract $8 \sqrt[3]{x + 4}$ from $- 8 \sqrt[3]{x + 4}$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 (\sqrt[3]{{(x + 4)}^{2}} - 16 \sqrt[3]{x + 4} + 64) + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.7

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 \sqrt[3]{{(x + 4)}^{2}} + 24 (- 16 \sqrt[3]{x + 4}) + 24 \cdot 64 + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.8

Simplify.

Tap for more steps...

Step 5.2.3.8.1

Multiply $- 16$ by $24$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 \sqrt[3]{{(x + 4)}^{2}} - 384 \sqrt[3]{x + 4} + 24 \cdot 64 + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.8.2

Multiply $24$ by $64$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 \sqrt[3]{{(x + 4)}^{2}} - 384 \sqrt[3]{x + 4} + 1536 + 192 (\sqrt[3]{x + 4} - 8) + 508$

Step 5.2.3.9

Apply the distributive property.

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 \sqrt[3]{{(x + 4)}^{2}} - 384 \sqrt[3]{x + 4} + 1536 + 192 \sqrt[3]{x + 4} + 192 \cdot - 8 + 508$

Step 5.2.3.10

Multiply $192$ by $- 8$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 \sqrt[3]{{(x + 4)}^{2}} - 384 \sqrt[3]{x + 4} + 1536 + 192 \sqrt[3]{x + 4} - 1536 + 508$

Step 5.2.4

Simplify by adding terms.

Tap for more steps...

Step 5.2.4.1

Combine the opposite terms in $x - 508 - 24 \sqrt[3]{{(x + 4)}^{2}} + 192 \sqrt[3]{x + 4} + 24 \sqrt[3]{{(x + 4)}^{2}} - 384 \sqrt[3]{x + 4} + 1536 + 192 \sqrt[3]{x + 4} - 1536 + 508$ .

Tap for more steps...

Step 5.2.4.1.1

Add $- 24 \sqrt[3]{{(x + 4)}^{2}}$ and $24 \sqrt[3]{{(x + 4)}^{2}}$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 + 0 + 192 \sqrt[3]{x + 4} - 384 \sqrt[3]{x + 4} + 1536 + 192 \sqrt[3]{x + 4} - 1536 + 508$

Step 5.2.4.1.2

Add $x - 508$ and $0$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 + 192 \sqrt[3]{x + 4} - 384 \sqrt[3]{x + 4} + 1536 + 192 \sqrt[3]{x + 4} - 1536 + 508$

Step 5.2.4.1.3

Subtract $1536$ from $1536$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 + 192 \sqrt[3]{x + 4} - 384 \sqrt[3]{x + 4} + 0 + 192 \sqrt[3]{x + 4} + 508$

Step 5.2.4.1.4

Add $x - 508 + 192 \sqrt[3]{x + 4} - 384 \sqrt[3]{x + 4}$ and $0$ .

$f^{- 1} (\sqrt[3]{x + 4} - 8) = x - 508 + 192 \sqrt[3]{x + 4} - 384 \sqrt[3]{x + 4} + 192 \sqrt[3]{x + 4} + 508$

Step 5.2.4.1.5

Add $- 508$ and $508$ .

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

Step 5.2.4.1.6

Add $x$ and $0$ .

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

Step 5.2.4.2

Subtract $384 \sqrt[3]{x + 4}$ from $192 \sqrt[3]{x + 4}$ .

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

Step 5.2.4.3

Combine the opposite terms in $x - 192 \sqrt[3]{x + 4} + 192 \sqrt[3]{x + 4}$ .

Tap for more steps...

Step 5.2.4.3.1

Add $- 192 \sqrt[3]{x + 4}$ and $192 \sqrt[3]{x + 4}$ .

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

Step 5.2.4.3.2

Add $x$ and $0$ .

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (x^{3} + 24 x^{2} + 192 x + 508)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x^{3} + 24 x^{2} + 192 x + 508) + 4} - 8$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

Add $508$ and $4$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{x^{3} + 24 x^{2} + 192 x + 512} - 8$

Step 5.3.3.2

Rewrite $x^{3} + 24 x^{2} + 192 x + 512$ in a factored form.

Tap for more steps...

Step 5.3.3.2.1

Regroup terms.

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{x^{3} + 512 + 24 x^{2} + 192 x} - 8$

Step 5.3.3.2.2

Rewrite $512$ as $8^{3}$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{x^{3} + 8^{3} + 24 x^{2} + 192 x} - 8$

Step 5.3.3.2.3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 8$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - x \cdot 8 + 8^{2}) + 24 x^{2} + 192 x} - 8$

Step 5.3.3.2.4

Simplify.

Tap for more steps...

Step 5.3.3.2.4.1

Multiply $8$ by $- 1$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 8^{2}) + 24 x^{2} + 192 x} - 8$

Step 5.3.3.2.4.2

Raise $8$ to the power of $2$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 64) + 24 x^{2} + 192 x} - 8$

Step 5.3.3.2.5

Factor $24 x$ out of $24 x^{2} + 192 x$ .

Tap for more steps...

Step 5.3.3.2.5.1

Factor $24 x$ out of $24 x^{2}$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 64) + 24 x (x) + 192 x} - 8$

Step 5.3.3.2.5.2

Factor $24 x$ out of $192 x$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 64) + 24 x (x) + 24 x (8)} - 8$

Step 5.3.3.2.5.3

Factor $24 x$ out of $24 x (x) + 24 x (8)$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 64) + 24 x (x + 8)} - 8$

Step 5.3.3.2.6

Factor $x + 8$ out of $(x + 8) (x^{2} - 8 x + 64) + 24 x (x + 8)$ .

Tap for more steps...

Step 5.3.3.2.6.1

Factor $x + 8$ out of $24 x (x + 8)$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 64) + (x + 8) (24 x)} - 8$

Step 5.3.3.2.6.2

Factor $x + 8$ out of $(x + 8) (x^{2} - 8 x + 64) + (x + 8) (24 x)$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} - 8 x + 64 + 24 x)} - 8$

Step 5.3.3.2.7

Add $- 8 x$ and $24 x$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} + 16 x + 64)} - 8$

Step 5.3.3.2.8

Factor using the perfect square rule.

Tap for more steps...

Step 5.3.3.2.8.1

Rewrite $64$ as $8^{2}$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} + 16 x + 8^{2})} - 8$

Step 5.3.3.2.8.2

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

$16 x = 2 \cdot x \cdot 8$

Step 5.3.3.2.8.3

Rewrite the polynomial.

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) (x^{2} + 2 \cdot x \cdot 8 + 8^{2})} - 8$

Step 5.3.3.2.8.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 8$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) {(x + 8)}^{2}} - 8$

Step 5.3.3.3

Multiply $x + 8$ by ${(x + 8)}^{2}$ by adding the exponents.

Tap for more steps...

Step 5.3.3.3.1

Multiply $x + 8$ by ${(x + 8)}^{2}$ .

Tap for more steps...

Step 5.3.3.3.1.1

Raise $x + 8$ to the power of $1$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{(x + 8) {(x + 8)}^{2}} - 8$

Step 5.3.3.3.1.2

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

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{{(x + 8)}^{1 + 2}} - 8$

Step 5.3.3.3.2

Add $1$ and $2$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = \sqrt[3]{{(x + 8)}^{3}} - 8$

Step 5.3.3.4

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

$f (x^{3} + 24 x^{2} + 192 x + 508) = x + 8 - 8$

Step 5.3.4

Combine the opposite terms in $x + 8 - 8$ .

Tap for more steps...

Step 5.3.4.1

Subtract $8$ from $8$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (x^{3} + 24 x^{2} + 192 x + 508) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{3} + 24 x^{2} + 192 x + 508$ is the inverse of $f (x) = \sqrt[3]{x + 4} - 8$ .

$f^{- 1} (x) = x^{3} + 24 x^{2} + 192 x + 508$