Algebra Examples

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Subtract $4$ from both sides of the equation.

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

Step 3.3

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

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

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

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

Step 3.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.2.1

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

Tap for more steps...

Step 3.4.2.1.1

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

Tap for more steps...

Step 3.4.2.1.1.1

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

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

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.

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

Step 3.4.2.1.1.2.2

Rewrite the expression.

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

Step 3.4.2.1.2

Simplify.

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

Step 3.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.3.1

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

Tap for more steps...

Step 3.4.3.1.1

Use the Binomial Theorem.

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

Step 3.4.3.1.2

Simplify each term.

Tap for more steps...

Step 3.4.3.1.2.1

Multiply $- 4$ by $3$ .

$8 y = x^{3} - 12 x^{2} + 3 x {(- 4)}^{2} + {(- 4)}^{3}$

Step 3.4.3.1.2.2

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

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

Step 3.4.3.1.2.3

Multiply $16$ by $3$ .

$8 y = x^{3} - 12 x^{2} + 48 x + {(- 4)}^{3}$

Step 3.4.3.1.2.4

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

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

Step 3.5

Divide each term in $8 y = x^{3} - 12 x^{2} + 48 x - 64$ by $8$ and simplify.

Tap for more steps...

Step 3.5.1

Divide each term in $8 y = x^{3} - 12 x^{2} + 48 x - 64$ by $8$ .

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

Step 3.5.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.1

Cancel the common factor of $8$ .

Tap for more steps...

Step 3.5.2.1.1

Cancel the common factor.

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

Step 3.5.2.1.2

Divide $y$ by $1$ .

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

Step 3.5.3

Simplify the right side.

Tap for more steps...

Step 3.5.3.1

Simplify each term.

Tap for more steps...

Step 3.5.3.1.1

Cancel the common factor of $- 12$ and $8$ .

Tap for more steps...

Step 3.5.3.1.1.1

Factor $4$ out of $- 12 x^{2}$ .

$y = \frac{x^{3}}{8} + \frac{4 (- 3 x^{2})}{8} + \frac{48 x}{8} + \frac{- 64}{8}$

Step 3.5.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 3.5.3.1.1.2.1

Factor $4$ out of $8$ .

$y = \frac{x^{3}}{8} + \frac{4 (- 3 x^{2})}{4 (2)} + \frac{48 x}{8} + \frac{- 64}{8}$

Step 3.5.3.1.1.2.2

Cancel the common factor.

$y = \frac{x^{3}}{8} + \frac{4 (- 3 x^{2})}{4 \cdot 2} + \frac{48 x}{8} + \frac{- 64}{8}$

Step 3.5.3.1.1.2.3

Rewrite the expression.

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

Step 3.5.3.1.2

Move the negative in front of the fraction.

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

Step 3.5.3.1.3

Cancel the common factor of $48$ and $8$ .

Tap for more steps...

Step 3.5.3.1.3.1

Factor $8$ out of $48 x$ .

$y = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + \frac{8 (6 x)}{8} + \frac{- 64}{8}$

Step 3.5.3.1.3.2

Cancel the common factors.

Tap for more steps...

Step 3.5.3.1.3.2.1

Factor $8$ out of $8$ .

$y = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + \frac{8 (6 x)}{8 (1)} + \frac{- 64}{8}$

Step 3.5.3.1.3.2.2

Cancel the common factor.

$y = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + \frac{8 (6 x)}{8 \cdot 1} + \frac{- 64}{8}$

Step 3.5.3.1.3.2.3

Rewrite the expression.

$y = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + \frac{6 x}{1} + \frac{- 64}{8}$

Step 3.5.3.1.3.2.4

Divide $6 x$ by $1$ .

$y = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x + \frac{- 64}{8}$

Step 3.5.3.1.4

Divide $- 64$ by $8$ .

$y = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8$

Step 4

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

$f^{- 1} (x) = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8$ is the inverse of $f (x) = \sqrt[3]{8 x} + 4$ .

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(\sqrt[3]{8 x} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3

Simplify each term.

Tap for more steps...

Step 5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 5.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 x}$ as ${(8 x)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{({(8 x)}^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.2

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(8^{\frac{1}{3}} x^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.3

Rewrite $8$ as $2^{3}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{({(2^{3})}^{\frac{1}{3}} x^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.4

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2^{3 (\frac{1}{3})} x^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.1.5.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2^{3 (\frac{1}{3})} x^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.5.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2 x^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.6

Evaluate the exponent.

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2 x^{\frac{1}{3}} + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.7

Factor $2$ out of $2 x^{\frac{1}{3}} + 4$ .

Tap for more steps...

Step 5.2.3.1.7.1

Factor $2$ out of $2 x^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2 (x^{\frac{1}{3}}) + 4)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.7.2

Factor $2$ out of $4$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2 x^{\frac{1}{3}} + 2 \cdot 2)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.7.3

Factor $2$ out of $2 x^{\frac{1}{3}} + 2 \cdot 2$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{{(2 (x^{\frac{1}{3}} + 2))}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.8

Apply the product rule to $2 (x^{\frac{1}{3}} + 2)$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{2^{3} {(x^{\frac{1}{3}} + 2)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.1.9

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{8 {(x^{\frac{1}{3}} + 2)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{8 x} + 4) = \frac{8 {(x^{\frac{1}{3}} + 2)}^{3}}{8} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.3

Divide ${(x^{\frac{1}{3}} + 2)}^{3}$ by $1$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = {(x^{\frac{1}{3}} + 2)}^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.4

Use the Binomial Theorem.

$f^{- 1} (\sqrt[3]{8 x} + 4) = {(x^{\frac{1}{3}})}^{3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5

Simplify each term.

Tap for more steps...

Step 5.2.3.5.1

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

Tap for more steps...

Step 5.2.3.5.1.1

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.5.1.2.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x^{\frac{1}{3} \cdot 3} + 3 {(x^{\frac{1}{3}})}^{2} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.1.2.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.2

Simplify.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 3 {(x^{\frac{1}{3}})}^{2} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

Tap for more steps...

Step 5.2.3.5.3.1

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 3 x^{\frac{1}{3} \cdot 2} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.3.2

Combine $\frac{1}{3}$ and $2$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 3 x^{\frac{2}{3}} \cdot 2 + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.4

Multiply $2$ by $3$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 2^{2} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.5

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 4 + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.6

Multiply $4$ by $3$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 2^{3} - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.5.7

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(\sqrt[3]{8 x} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6

Simplify the numerator.

Tap for more steps...

Step 5.2.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 x}$ as ${(8 x)}^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {({(8 x)}^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.2

Apply the product rule to $8 x$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(8^{\frac{1}{3}} x^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.3

Rewrite $8$ as $2^{3}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {({(2^{3})}^{\frac{1}{3}} x^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.4

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2^{3 (\frac{1}{3})} x^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.3.6.5.1

Cancel the common factor.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2^{3 (\frac{1}{3})} x^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.5.2

Rewrite the expression.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2 x^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.6

Evaluate the exponent.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2 x^{\frac{1}{3}} + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.7

Factor $2$ out of $2 x^{\frac{1}{3}} + 4$ .

Tap for more steps...

Step 5.2.3.6.7.1

Factor $2$ out of $2 x^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2 (x^{\frac{1}{3}}) + 4)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.7.2

Factor $2$ out of $4$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2 x^{\frac{1}{3}} + 2 \cdot 2)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.7.3

Factor $2$ out of $2 x^{\frac{1}{3}} + 2 \cdot 2$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 {(2 (x^{\frac{1}{3}} + 2))}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.8

Apply the product rule to $2 (x^{\frac{1}{3}} + 2)$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 \cdot (2^{2} {(x^{\frac{1}{3}} + 2)}^{2})}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.9

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{3 \cdot (4 {(x^{\frac{1}{3}} + 2)}^{2})}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.6.10

Multiply $3$ by $4$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{12 {(x^{\frac{1}{3}} + 2)}^{2}}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.7

Factor $2$ out of $12 {(x^{\frac{1}{3}} + 2)}^{2}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{2 (6 {(x^{\frac{1}{3}} + 2)}^{2})}{2} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.8

Cancel the common factors.

Tap for more steps...

Step 5.2.3.8.1

Factor $2$ out of $2$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{2 (6 {(x^{\frac{1}{3}} + 2)}^{2})}{2 (1)} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.8.2

Cancel the common factor.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{2 (6 {(x^{\frac{1}{3}} + 2)}^{2})}{2 \cdot 1} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.8.3

Rewrite the expression.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - \frac{6 {(x^{\frac{1}{3}} + 2)}^{2}}{1} + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.8.4

Divide $6 {(x^{\frac{1}{3}} + 2)}^{2}$ by $1$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 {(x^{\frac{1}{3}} + 2)}^{2}) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.9

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 ((x^{\frac{1}{3}} + 2) (x^{\frac{1}{3}} + 2))) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.10

Expand $(x^{\frac{1}{3}} + 2) (x^{\frac{1}{3}} + 2)$ using the FOIL Method.

Tap for more steps...

Step 5.2.3.10.1

Apply the distributive property.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{1}{3}} (x^{\frac{1}{3}} + 2) + 2 (x^{\frac{1}{3}} + 2))) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.10.2

Apply the distributive property.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 2 + 2 (x^{\frac{1}{3}} + 2))) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.10.3

Apply the distributive property.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{1}{3}} x^{\frac{1}{3}} + x^{\frac{1}{3}} \cdot 2 + 2 x^{\frac{1}{3}} + 2 \cdot 2)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.11

Simplify and combine like terms.

Tap for more steps...

Step 5.2.3.11.1

Simplify each term.

Tap for more steps...

Step 5.2.3.11.1.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{1}{3}}$ by adding the exponents.

Tap for more steps...

Step 5.2.3.11.1.1.1

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

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{1}{3} + \frac{1}{3}} + x^{\frac{1}{3}} \cdot 2 + 2 x^{\frac{1}{3}} + 2 \cdot 2)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.11.1.1.2

Combine the numerators over the common denominator.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{1 + 1}{3}} + x^{\frac{1}{3}} \cdot 2 + 2 x^{\frac{1}{3}} + 2 \cdot 2)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.11.1.1.3

Add $1$ and $1$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot 2 + 2 x^{\frac{1}{3}} + 2 \cdot 2)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.11.1.2

Move $2$ to the left of $x^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{2}{3}} + 2 \cdot x^{\frac{1}{3}} + 2 x^{\frac{1}{3}} + 2 \cdot 2)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.11.1.3

Multiply $2$ by $2$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{2}{3}} + 2 x^{\frac{1}{3}} + 2 x^{\frac{1}{3}} + 4)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.11.2

Add $2 x^{\frac{1}{3}}$ and $2 x^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 (x^{\frac{2}{3}} + 4 x^{\frac{1}{3}} + 4)) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.12

Apply the distributive property.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 x^{\frac{2}{3}} + 6 (4 x^{\frac{1}{3}}) + 6 \cdot 4) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.13

Simplify.

Tap for more steps...

Step 5.2.3.13.1

Multiply $4$ by $6$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 x^{\frac{2}{3}} + 24 x^{\frac{1}{3}} + 6 \cdot 4) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.13.2

Multiply $6$ by $4$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 x^{\frac{2}{3}} + 24 x^{\frac{1}{3}} + 24) + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.14

Apply the distributive property.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - (6 x^{\frac{2}{3}}) - (24 x^{\frac{1}{3}}) - 1 \cdot 24 + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.15

Simplify.

Tap for more steps...

Step 5.2.3.15.1

Multiply $6$ by $- 1$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - (24 x^{\frac{1}{3}}) - 1 \cdot 24 + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.15.2

Multiply $24$ by $- 1$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 1 \cdot 24 + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.15.3

Multiply $- 1$ by $24$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 6 (\sqrt[3]{8 x} + 4) - 8$

Step 5.2.3.16

Simplify each term.

Tap for more steps...

Step 5.2.3.16.1

Rewrite $8$ as $2^{3}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 6 (\sqrt[3]{2^{3} x} + 4) - 8$

Step 5.2.3.16.2

Pull terms out from under the radical.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 6 (2 \sqrt[3]{x} + 4) - 8$

Step 5.2.3.17

Apply the distributive property.

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 6 (2 \sqrt[3]{x}) + 6 \cdot 4 - 8$

Step 5.2.3.18

Multiply $2$ by $6$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 12 \sqrt[3]{x} + 6 \cdot 4 - 8$

Step 5.2.3.19

Multiply $6$ by $4$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 12 \sqrt[3]{x} + 24 - 8$

Step 5.2.4

Simplify by adding terms.

Tap for more steps...

Step 5.2.4.1

Combine the opposite terms in $x + 6 x^{\frac{2}{3}} + 12 x^{\frac{1}{3}} + 8 - 6 x^{\frac{2}{3}} - 24 x^{\frac{1}{3}} - 24 + 12 \sqrt[3]{x} + 24 - 8$ .

Tap for more steps...

Step 5.2.4.1.1

Subtract $6 x^{\frac{2}{3}}$ from $6 x^{\frac{2}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 12 x^{\frac{1}{3}} + 8 + 0 - 24 x^{\frac{1}{3}} - 24 + 12 \sqrt[3]{x} + 24 - 8$

Step 5.2.4.1.2

Add $x + 12 x^{\frac{1}{3}} + 8$ and $0$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 12 x^{\frac{1}{3}} + 8 - 24 x^{\frac{1}{3}} - 24 + 12 \sqrt[3]{x} + 24 - 8$

Step 5.2.4.1.3

Add $- 24$ and $24$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 12 x^{\frac{1}{3}} + 8 - 24 x^{\frac{1}{3}} + 0 + 12 \sqrt[3]{x} - 8$

Step 5.2.4.1.4

Add $x + 12 x^{\frac{1}{3}} + 8 - 24 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 12 x^{\frac{1}{3}} + 8 - 24 x^{\frac{1}{3}} + 12 \sqrt[3]{x} - 8$

Step 5.2.4.1.5

Subtract $8$ from $8$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 12 x^{\frac{1}{3}} + 0 - 24 x^{\frac{1}{3}} + 12 \sqrt[3]{x}$

Step 5.2.4.1.6

Add $x + 12 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x + 12 x^{\frac{1}{3}} - 24 x^{\frac{1}{3}} + 12 \sqrt[3]{x}$

Step 5.2.4.2

Subtract $24 x^{\frac{1}{3}}$ from $12 x^{\frac{1}{3}}$ .

$f^{- 1} (\sqrt[3]{8 x} + 4) = x - 12 x^{\frac{1}{3}} + 12 \sqrt[3]{x}$

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 $f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8)} + 4$

Step 5.3.3

Simplify each term.

Tap for more steps...

Step 5.3.3.1

To write $- \frac{3 x^{2}}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} \cdot \frac{4}{4} + 6 x - 8)} + 4$

Step 5.3.3.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.3.3.2.1

Multiply $\frac{3 x^{2}}{2}$ by $\frac{4}{4}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3}}{8} - \frac{3 x^{2} \cdot 4}{2 \cdot 4} + 6 x - 8)} + 4$

Step 5.3.3.2.2

Multiply $2$ by $4$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3}}{8} - \frac{3 x^{2} \cdot 4}{8} + 6 x - 8)} + 4$

Step 5.3.3.3

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 3 x^{2} \cdot 4}{8} + 6 x - 8)} + 4$

Step 5.3.3.4

Simplify the numerator.

Tap for more steps...

Step 5.3.3.4.1

Factor $x^{2}$ out of $x^{3} - 3 x^{2} \cdot 4$ .

Tap for more steps...

Step 5.3.3.4.1.1

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

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} x - 3 x^{2} \cdot 4}{8} + 6 x - 8)} + 4$

Step 5.3.3.4.1.2

Factor $x^{2}$ out of $- 3 x^{2} \cdot 4$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} x + x^{2} (- 3 \cdot 4)}{8} + 6 x - 8)} + 4$

Step 5.3.3.4.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} (- 3 \cdot 4)$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} (x - 3 \cdot 4)}{8} + 6 x - 8)} + 4$

Step 5.3.3.4.2

Multiply $- 3$ by $4$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} (x - 12)}{8} + 6 x - 8)} + 4$

Step 5.3.3.5

To write $6 x$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} (x - 12)}{8} + 6 x \cdot \frac{8}{8} - 8)} + 4$

Step 5.3.3.6

Combine $6 x$ and $\frac{8}{8}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} (x - 12)}{8} + \frac{6 x \cdot 8}{8} - 8)} + 4$

Step 5.3.3.7

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{2} (x - 12) + 6 x \cdot 8}{8} - 8)} + 4$

Step 5.3.3.8

Simplify the numerator.

Tap for more steps...

Step 5.3.3.8.1

Factor $x$ out of $x^{2} (x - 12) + 6 x \cdot 8$ .

Tap for more steps...

Step 5.3.3.8.1.1

Factor $x$ out of $x^{2} (x - 12)$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x (x - 12)) + 6 x \cdot 8}{8} - 8)} + 4$

Step 5.3.3.8.1.2

Factor $x$ out of $6 x \cdot 8$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x (x - 12)) + x (6 \cdot 8)}{8} - 8)} + 4$

Step 5.3.3.8.1.3

Factor $x$ out of $x (x (x - 12)) + x (6 \cdot 8)$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x (x - 12) + 6 \cdot 8)}{8} - 8)} + 4$

Step 5.3.3.8.2

Apply the distributive property.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x \cdot x + x \cdot - 12 + 6 \cdot 8)}{8} - 8)} + 4$

Step 5.3.3.8.3

Multiply $x$ by $x$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x^{2} + x \cdot - 12 + 6 \cdot 8)}{8} - 8)} + 4$

Step 5.3.3.8.4

Move $- 12$ to the left of $x$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x^{2} - 12 \cdot x + 6 \cdot 8)}{8} - 8)} + 4$

Step 5.3.3.8.5

Multiply $6$ by $8$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x^{2} - 12 x + 48)}{8} - 8)} + 4$

Step 5.3.3.9

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x^{2} - 12 x + 48)}{8} - 8 \cdot \frac{8}{8})} + 4$

Step 5.3.3.10

Combine $- 8$ and $\frac{8}{8}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x^{2} - 12 x + 48)}{8} + \frac{- 8 \cdot 8}{8})} + 4$

Step 5.3.3.11

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x (x^{2} - 12 x + 48) - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12

Simplify the numerator.

Tap for more steps...

Step 5.3.3.12.1

Apply the distributive property.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x \cdot x^{2} + x (- 12 x) + x \cdot 48 - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.2

Simplify.

Tap for more steps...

Step 5.3.3.12.2.1

Multiply $x$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 5.3.3.12.2.1.1

Multiply $x$ by $x^{2}$ .

Tap for more steps...

Step 5.3.3.12.2.1.1.1

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

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x \cdot x^{2} + x (- 12 x) + x \cdot 48 - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.2.1.1.2

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

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{1 + 2} + x (- 12 x) + x \cdot 48 - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.2.1.2

Add $1$ and $2$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} + x (- 12 x) + x \cdot 48 - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.2.2

Rewrite using the commutative property of multiplication.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 12 x \cdot x + x \cdot 48 - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.2.3

Move $48$ to the left of $x$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 12 x \cdot x + 48 \cdot x - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 5.3.3.12.3.1

Move $x$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 12 (x \cdot x) + 48 \cdot x - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.3.2

Multiply $x$ by $x$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 12 x^{2} + 48 \cdot x - 8 \cdot 8}{8})} + 4$

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 12 x^{2} + 48 x - 8 \cdot 8}{8})} + 4$

Step 5.3.3.12.4

Multiply $- 8$ by $8$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{x^{3} - 12 x^{2} + 48 x - 64}{8})} + 4$

Step 5.3.3.12.5

Rewrite $x^{3} - 12 x^{2} + 48 x - 64$ in a factored form.

Tap for more steps...

Step 5.3.3.12.5.1

Factor $x^{3} - 12 x^{2} + 48 x - 64$ using the rational roots test.

Tap for more steps...

Step 5.3.3.12.5.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 64, \pm 2, \pm 32, \pm 4, \pm 16, \pm 8$

$q = \pm 1$

Step 5.3.3.12.5.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 64, \pm 2, \pm 32, \pm 4, \pm 16, \pm 8$

Step 5.3.3.12.5.1.3

Substitute $4$ and simplify the expression. In this case, the expression is equal to $0$ so $4$ is a root of the polynomial.

Tap for more steps...

Step 5.3.3.12.5.1.3.1

Substitute $4$ into the polynomial.

$4^{3} - 12 \cdot 4^{2} + 48 \cdot 4 - 64$

Step 5.3.3.12.5.1.3.2

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

$64 - 12 \cdot 4^{2} + 48 \cdot 4 - 64$

Step 5.3.3.12.5.1.3.3

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

$64 - 12 \cdot 16 + 48 \cdot 4 - 64$

Step 5.3.3.12.5.1.3.4

Multiply $- 12$ by $16$ .

$64 - 192 + 48 \cdot 4 - 64$

Step 5.3.3.12.5.1.3.5

Subtract $192$ from $64$ .

$- 128 + 48 \cdot 4 - 64$

Step 5.3.3.12.5.1.3.6

Multiply $48$ by $4$ .

$- 128 + 192 - 64$

Step 5.3.3.12.5.1.3.7

Add $- 128$ and $192$ .

$64 - 64$

Step 5.3.3.12.5.1.3.8

Subtract $64$ from $64$ .

$0$

Step 5.3.3.12.5.1.4

Since $4$ is a known root, divide the polynomial by $x - 4$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 12 x^{2} + 48 x - 64}{x - 4}$

Step 5.3.3.12.5.1.5

Divide $x^{3} - 12 x^{2} + 48 x - 64$ by $x - 4$ .

Tap for more steps...

Step 5.3.3.12.5.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$4$

$x^{3}$

$12 x^{2}$

$48 x$

$64$

Step 5.3.3.12.5.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$

Step 5.3.3.12.5.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			+	$x^{3}$	-	$4 x^{2}$

Step 5.3.3.12.5.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 4 x^{2}$

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$

Step 5.3.3.12.5.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$

Step 5.3.3.12.5.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$

Step 5.3.3.12.5.1.5.7

Divide the highest order term in the dividend $- 8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$

Step 5.3.3.12.5.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					-	$8 x^{2}$	+	$32 x$

Step 5.3.3.12.5.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{2} + 32 x$

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$

Step 5.3.3.12.5.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$
							+	$16 x$

Step 5.3.3.12.5.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	-	$8 x$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$
							+	$16 x$	-	$64$

Step 5.3.3.12.5.1.5.12

Divide the highest order term in the dividend $16 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$
							+	$16 x$	-	$64$

Step 5.3.3.12.5.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$
							+	$16 x$	-	$64$
							+	$16 x$	-	$64$

Step 5.3.3.12.5.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $16 x - 64$

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$
							+	$16 x$	-	$64$
							-	$16 x$	+	$64$

Step 5.3.3.12.5.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	-	$8 x$	+	$16$
$x$	-	$4$		$x^{3}$	-	$12 x^{2}$	+	$48 x$	-	$64$
			-	$x^{3}$	+	$4 x^{2}$
					-	$8 x^{2}$	+	$48 x$
					+	$8 x^{2}$	-	$32 x$
							+	$16 x$	-	$64$
							-	$16 x$	+	$64$
										$0$

Step 5.3.3.12.5.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} - 8 x + 16$

Step 5.3.3.12.5.1.6

Write $x^{3} - 12 x^{2} + 48 x - 64$ as a set of factors.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{(x - 4) (x^{2} - 8 x + 16)}{8})} + 4$

Step 5.3.3.12.5.2

Factor using the perfect square rule.

Tap for more steps...

Step 5.3.3.12.5.2.1

Rewrite $16$ as $4^{2}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{(x - 4) (x^{2} - 8 x + 4^{2})}{8})} + 4$

Step 5.3.3.12.5.2.2

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

$8 x = 2 \cdot x \cdot 4$

Step 5.3.3.12.5.2.3

Rewrite the polynomial.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{(x - 4) (x^{2} - 2 \cdot x \cdot 4 + 4^{2})}{8})} + 4$

Step 5.3.3.12.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{(x - 4) {(x - 4)}^{2}}{8})} + 4$

Step 5.3.3.12.5.3

Combine like factors.

Tap for more steps...

Step 5.3.3.12.5.3.1

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

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{(x - 4) {(x - 4)}^{2}}{8})} + 4$

Step 5.3.3.12.5.3.2

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

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{{(x - 4)}^{1 + 2}}{8})} + 4$

Step 5.3.3.12.5.3.3

Add $1$ and $2$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{8 (\frac{{(x - 4)}^{3}}{8})} + 4$

Step 5.3.3.13

Combine $8$ and $\frac{{(x - 4)}^{3}}{8}$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{\frac{8 {(x - 4)}^{3}}{8}} + 4$

Step 5.3.3.14

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 5.3.3.14.1

Reduce the expression $\frac{8 {(x - 4)}^{3}}{8}$ by cancelling the common factors.

Tap for more steps...

Step 5.3.3.14.1.1

Cancel the common factor.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{\frac{8 {(x - 4)}^{3}}{8}} + 4$

Step 5.3.3.14.1.2

Rewrite the expression.

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{\frac{{(x - 4)}^{3}}{1}} + 4$

Step 5.3.3.14.2

Divide ${(x - 4)}^{3}$ by $1$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = \sqrt[3]{{(x - 4)}^{3}} + 4$

Step 5.3.3.15

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

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = x - 4 + 4$

Step 5.3.4

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

Tap for more steps...

Step 5.3.4.1

Add $- 4$ and $4$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (\frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8$ is the inverse of $f (x) = \sqrt[3]{8 x} + 4$ .

$f^{- 1} (x) = \frac{x^{3}}{8} - \frac{3 x^{2}}{2} + 6 x - 8$