Algebra Examples

$- \sqrt[3]{3 x - 6} + 12$

Step 1

Interchange the variables.

$x = - \sqrt[3]{3 y - 6} + 12$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $- \sqrt[3]{3 y - 6} + 12 = x$ .

$- \sqrt[3]{3 y - 6} + 12 = x$

Step 2.2

Subtract $12$ from both sides of the equation.

$- \sqrt[3]{3 y - 6} = x - 12$

Step 2.3

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

${(- \sqrt[3]{3 y - 6})}^{3} = {(x - 12)}^{3}$

Step 2.4

Simplify each side of the equation.

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Step 2.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 y - 6}$ as ${(3 y - 6)}^{\frac{1}{3}}$ .

${(- {(3 y - 6)}^{\frac{1}{3}})}^{3} = {(x - 12)}^{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Simplify ${(- {(3 y - 6)}^{\frac{1}{3}})}^{3}$ .

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Step 2.4.2.1.1

Apply the product rule to $- {(3 y - 6)}^{\frac{1}{3}}$ .

${(- 1)}^{3} {({(3 y - 6)}^{\frac{1}{3}})}^{3} = {(x - 12)}^{3}$

Step 2.4.2.1.2

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

$- {({(3 y - 6)}^{\frac{1}{3}})}^{3} = {(x - 12)}^{3}$

Step 2.4.2.1.3

Multiply the exponents in ${({(3 y - 6)}^{\frac{1}{3}})}^{3}$ .

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Step 2.4.2.1.3.1

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

$- {(3 y - 6)}^{\frac{1}{3} \cdot 3} = {(x - 12)}^{3}$

Step 2.4.2.1.3.2

Cancel the common factor of $3$ .

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Step 2.4.2.1.3.2.1

Cancel the common factor.

$- {(3 y - 6)}^{\frac{1}{3} \cdot 3} = {(x - 12)}^{3}$

Step 2.4.2.1.3.2.2

Rewrite the expression.

$- {(3 y - 6)}^{1} = {(x - 12)}^{3}$

Step 2.4.2.1.4

Simplify.

$- (3 y - 6) = {(x - 12)}^{3}$

Step 2.4.2.1.5

Apply the distributive property.

$- (3 y) - - 6 = {(x - 12)}^{3}$

Step 2.4.2.1.6

Multiply.

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Step 2.4.2.1.6.1

Multiply $3$ by $- 1$ .

$- 3 y - - 6 = {(x - 12)}^{3}$

Step 2.4.2.1.6.2

Multiply $- 1$ by $- 6$ .

$- 3 y + 6 = {(x - 12)}^{3}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Simplify ${(x - 12)}^{3}$ .

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Step 2.4.3.1.1

Use the Binomial Theorem.

$- 3 y + 6 = x^{3} + 3 x^{2} \cdot - 12 + 3 x {(- 12)}^{2} + {(- 12)}^{3}$

Step 2.4.3.1.2

Simplify each term.

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Step 2.4.3.1.2.1

Multiply $- 12$ by $3$ .

$- 3 y + 6 = x^{3} - 36 x^{2} + 3 x {(- 12)}^{2} + {(- 12)}^{3}$

Step 2.4.3.1.2.2

Raise $- 12$ to the power of $2$ .

$- 3 y + 6 = x^{3} - 36 x^{2} + 3 x \cdot 144 + {(- 12)}^{3}$

Step 2.4.3.1.2.3

Multiply $144$ by $3$ .

$- 3 y + 6 = x^{3} - 36 x^{2} + 432 x + {(- 12)}^{3}$

Step 2.4.3.1.2.4

Raise $- 12$ to the power of $3$ .

$- 3 y + 6 = x^{3} - 36 x^{2} + 432 x - 1728$

Step 2.5

Solve for $y$ .

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Step 2.5.1

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

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Step 2.5.1.1

Subtract $6$ from both sides of the equation.

$- 3 y = x^{3} - 36 x^{2} + 432 x - 1728 - 6$

Step 2.5.1.2

Subtract $6$ from $- 1728$ .

$- 3 y = x^{3} - 36 x^{2} + 432 x - 1734$

Step 2.5.2

Divide each term in $- 3 y = x^{3} - 36 x^{2} + 432 x - 1734$ by $- 3$ and simplify.

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Step 2.5.2.1

Divide each term in $- 3 y = x^{3} - 36 x^{2} + 432 x - 1734$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{x^{3}}{- 3} + \frac{- 36 x^{2}}{- 3} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Cancel the common factor of $- 3$ .

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Step 2.5.2.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{x^{3}}{- 3} + \frac{- 36 x^{2}}{- 3} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{x^{3}}{- 3} + \frac{- 36 x^{2}}{- 3} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Simplify each term.

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Step 2.5.2.3.1.1

Move the negative in front of the fraction.

$y = - \frac{x^{3}}{3} + \frac{- 36 x^{2}}{- 3} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.2

Cancel the common factor of $- 36$ and $- 3$ .

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Step 2.5.2.3.1.2.1

Factor $- 3$ out of $- 36 x^{2}$ .

$y = - \frac{x^{3}}{3} + \frac{- 3 (12 x^{2})}{- 3} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.2.2

Cancel the common factors.

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Step 2.5.2.3.1.2.2.1

Factor $- 3$ out of $- 3$ .

$y = - \frac{x^{3}}{3} + \frac{- 3 (12 x^{2})}{- 3 (1)} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.2.2.2

Cancel the common factor.

$y = - \frac{x^{3}}{3} + \frac{- 3 (12 x^{2})}{- 3 \cdot 1} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.2.2.3

Rewrite the expression.

$y = - \frac{x^{3}}{3} + \frac{12 x^{2}}{1} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.2.2.4

Divide $12 x^{2}$ by $1$ .

$y = - \frac{x^{3}}{3} + 12 x^{2} + \frac{432 x}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.3

Cancel the common factor of $432$ and $- 3$ .

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Step 2.5.2.3.1.3.1

Factor $3$ out of $432 x$ .

$y = - \frac{x^{3}}{3} + 12 x^{2} + \frac{3 (144 x)}{- 3} + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.3.2

Move the negative one from the denominator of $\frac{144 x}{- 1}$ .

$y = - \frac{x^{3}}{3} + 12 x^{2} - 1 \cdot (144 x) + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.4

Rewrite $- 1 \cdot (144 x)$ as $- (144 x)$ .

$y = - \frac{x^{3}}{3} + 12 x^{2} - (144 x) + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.5

Multiply $144$ by $- 1$ .

$y = - \frac{x^{3}}{3} + 12 x^{2} - 144 x + \frac{- 1734}{- 3}$

Step 2.5.2.3.1.6

Divide $- 1734$ by $- 3$ .

$y = - \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578$

Step 3

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

$f^{- 1} (x) = - \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578$

Step 4

Verify if $f^{- 1} (x) = - \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578$ is the inverse of $f (x) = - \sqrt[3]{3 x - 6} + 12$ .

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

Evaluate $f^{- 1} (- \sqrt[3]{3 x - 6} + 12)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{{(- \sqrt[3]{3 x - 6} + 12)}^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Factor $3$ out of $3 x - 6$ .

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Step 4.2.3.1.1

Factor $3$ out of $3 x$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{{(- \sqrt[3]{3 (x) - 6} + 12)}^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.1.2

Factor $3$ out of $- 6$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{{(- \sqrt[3]{3 x + 3 \cdot - 2} + 12)}^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.1.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{{(- \sqrt[3]{3 (x - 2)} + 12)}^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2

Simplify ${(- \sqrt[3]{3 (x - 2)} + 12)}^{3}$ .

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Step 4.2.3.2.1

Use the Binomial Theorem.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{{(- \sqrt[3]{3 (x - 2)})}^{3} + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2

Simplify each term.

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Step 4.2.3.2.2.1

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{{(- 1)}^{3} {\sqrt[3]{3 (x - 2)}}^{3} + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.2

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- {\sqrt[3]{3 (x - 2)}}^{3} + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.3

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

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Step 4.2.3.2.2.3.1

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- {({(3 (x - 2))}^{\frac{1}{3}})}^{3} + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.3.2

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- {(3 (x - 2))}^{\frac{1}{3} \cdot 3} + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.3.3

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- {(3 (x - 2))}^{\frac{3}{3}} + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.3.4

Cancel the common factor of $3$ .

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Step 4.2.3.2.2.3.4.1

Cancel the common factor.

Step 4.2.3.2.2.3.4.2

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- (3 (x - 2)) + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.3.5

Simplify.

Step 4.2.3.2.2.4

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- (3 x + 3 \cdot - 2) + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.5

Multiply $3$ by $- 2$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- (3 x - 6) + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.6

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- (3 x) + 6 + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.7

Multiply $3$ by $- 1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 {(- \sqrt[3]{3 (x - 2)})}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.8

Multiply $- 1$ by $- 6$ .

Step 4.2.3.2.2.9

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 ({(- 1)}^{2} {\sqrt[3]{3 (x - 2)}}^{2}) \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.10

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 (1 {\sqrt[3]{3 (x - 2)}}^{2}) \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.11

Multiply ${\sqrt[3]{3 (x - 2)}}^{2}$ by $1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 {\sqrt[3]{3 (x - 2)}}^{2} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.12

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 \sqrt[3]{{(3 (x - 2))}^{2}} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.13

Apply the product rule to $3 (x - 2)$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 \sqrt[3]{3^{2} {(x - 2)}^{2}} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.14

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 3 \sqrt[3]{9 {(x - 2)}^{2}} \cdot 12 + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.15

Multiply $12$ by $3$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 36 \sqrt[3]{9 {(x - 2)}^{2}} + 3 (- \sqrt[3]{3 (x - 2)}) \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.16

Multiply $- 1$ by $3$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 3 \sqrt[3]{3 (x - 2)} \cdot 12^{2} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.17

Raise $12$ to the power of $2$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 3 \sqrt[3]{3 (x - 2)} \cdot 144 + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.18

Multiply $144$ by $- 3$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)} + 12^{3}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.2.19

Raise $12$ to the power of $3$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 6 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)} + 1728}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.2.3

Add $6$ and $1728$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- 3 x + 1734 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3

Cancel the common factor of $- 3 x + 1734 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)}$ and $3$ .

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Step 4.2.3.3.1

Factor $3$ out of $- 3 x$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x) + 1734 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.2

Factor $3$ out of $1734$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x) + 3 \cdot 578 + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.3

Factor $3$ out of $3 (- x) + 3 (578)$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578) + 36 \sqrt[3]{9 {(x - 2)}^{2}} - 432 \sqrt[3]{3 (x - 2)}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.4

Factor $3$ out of $36 \sqrt[3]{9 {(x - 2)}^{2}}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578) + 3 (12 \sqrt[3]{9 {(x - 2)}^{2}}) - 432 \sqrt[3]{3 (x - 2)}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.5

Factor $3$ out of $3 (- x + 578) + 3 (12 \sqrt[3]{9 {(x - 2)}^{2}})$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}}) - 432 \sqrt[3]{3 (x - 2)}}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.6

Factor $3$ out of $- 432 \sqrt[3]{3 (x - 2)}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}}) + 3 (- 144 \sqrt[3]{3 (x - 2)})}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.7

Factor $3$ out of $3 (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}}) + 3 (- 144 \sqrt[3]{3 (x - 2)})$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 144 \sqrt[3]{3 (x - 2)})}{3} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.8

Cancel the common factors.

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Step 4.2.3.3.8.1

Factor $3$ out of $3$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 144 \sqrt[3]{3 (x - 2)})}{3 (1)} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.8.2

Cancel the common factor.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{3 (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 144 \sqrt[3]{3 (x - 2)})}{3 \cdot 1} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.8.3

Rewrite the expression.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - \frac{- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 144 \sqrt[3]{3 (x - 2)}}{1} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.3.8.4

Divide $- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 144 \sqrt[3]{3 (x - 2)}$ by $1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = - (- x + 578 + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 144 \sqrt[3]{3 (x - 2)}) + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.4

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 1 \cdot 578 - (12 \sqrt[3]{9 {(x - 2)}^{2}}) - (- 144 \sqrt[3]{3 (x - 2)}) + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.5

Simplify.

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Step 4.2.3.5.1

Multiply $- - x$ .

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Step 4.2.3.5.1.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = 1 x - 1 \cdot 578 - (12 \sqrt[3]{9 {(x - 2)}^{2}}) - (- 144 \sqrt[3]{3 (x - 2)}) + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.5.1.2

Multiply $x$ by $1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 1 \cdot 578 - (12 \sqrt[3]{9 {(x - 2)}^{2}}) - (- 144 \sqrt[3]{3 (x - 2)}) + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.5.2

Multiply $- 1$ by $578$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - (12 \sqrt[3]{9 {(x - 2)}^{2}}) - (- 144 \sqrt[3]{3 (x - 2)}) + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.5.3

Multiply $12$ by $- 1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} - (- 144 \sqrt[3]{3 (x - 2)}) + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.5.4

Multiply $- 144$ by $- 1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 {(- \sqrt[3]{3 x - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.6

Factor $3$ out of $3 x - 6$ .

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Step 4.2.3.6.1

Factor $3$ out of $3 x$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 {(- \sqrt[3]{3 (x) - 6} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.6.2

Factor $3$ out of $- 6$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 {(- \sqrt[3]{3 x + 3 \cdot - 2} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.6.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 {(- \sqrt[3]{3 (x - 2)} + 12)}^{2} - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.7

Rewrite ${(- \sqrt[3]{3 (x - 2)} + 12)}^{2}$ as $(- \sqrt[3]{3 (x - 2)} + 12) (- \sqrt[3]{3 (x - 2)} + 12)$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 ((- \sqrt[3]{3 (x - 2)} + 12) (- \sqrt[3]{3 (x - 2)} + 12)) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.8

Expand $(- \sqrt[3]{3 (x - 2)} + 12) (- \sqrt[3]{3 (x - 2)} + 12)$ using the FOIL Method.

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Step 4.2.3.8.1

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (- \sqrt[3]{3 (x - 2)} (- \sqrt[3]{3 (x - 2)} + 12) + 12 (- \sqrt[3]{3 (x - 2)} + 12)) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.8.2

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (- \sqrt[3]{3 (x - 2)} (- \sqrt[3]{3 (x - 2)}) - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)} + 12)) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.8.3

Apply the distributive property.

Step 4.2.3.9

Simplify and combine like terms.

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Step 4.2.3.9.1

Simplify each term.

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Step 4.2.3.9.1.1

Multiply $- \sqrt[3]{3 (x - 2)} (- \sqrt[3]{3 (x - 2)})$ .

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Step 4.2.3.9.1.1.1

Multiply $- 1$ by $- 1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (1 \sqrt[3]{3 (x - 2)} \sqrt[3]{3 (x - 2)} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.1.2

Multiply $\sqrt[3]{3 (x - 2)}$ by $1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{3 (x - 2)} \sqrt[3]{3 (x - 2)} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.1.3

Raise $\sqrt[3]{3 (x - 2)}$ to the power of $1$ .

Step 4.2.3.9.1.1.4

Raise $\sqrt[3]{3 (x - 2)}$ to the power of $1$ .

Step 4.2.3.9.1.1.5

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 ({\sqrt[3]{3 (x - 2)}}^{1 + 1} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.1.6

Add $1$ and $1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 ({\sqrt[3]{3 (x - 2)}}^{2} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.2

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{{(3 (x - 2))}^{2}} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.3

Apply the product rule to $3 (x - 2)$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{3^{2} {(x - 2)}^{2}} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.4

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

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{9 {(x - 2)}^{2}} - \sqrt[3]{3 (x - 2)} \cdot 12 + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.5

Multiply $12$ by $- 1$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{9 {(x - 2)}^{2}} - 12 \sqrt[3]{3 (x - 2)} + 12 (- \sqrt[3]{3 (x - 2)}) + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.6

Multiply $- 1$ by $12$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{9 {(x - 2)}^{2}} - 12 \sqrt[3]{3 (x - 2)} - 12 \sqrt[3]{3 (x - 2)} + 12 \cdot 12) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.1.7

Multiply $12$ by $12$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{9 {(x - 2)}^{2}} - 12 \sqrt[3]{3 (x - 2)} - 12 \sqrt[3]{3 (x - 2)} + 144) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.9.2

Subtract $12 \sqrt[3]{3 (x - 2)}$ from $- 12 \sqrt[3]{3 (x - 2)}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 (\sqrt[3]{9 {(x - 2)}^{2}} - 24 \sqrt[3]{3 (x - 2)} + 144) - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.10

Apply the distributive property.

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} + 12 (- 24 \sqrt[3]{3 (x - 2)}) + 12 \cdot 144 - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.11

Simplify.

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Step 4.2.3.11.1

Multiply $- 24$ by $12$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 12 \cdot 144 - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.11.2

Multiply $12$ by $144$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 1728 - 144 (- \sqrt[3]{3 x - 6} + 12) + 578$

Step 4.2.3.12

Factor $3$ out of $3 x - 6$ .

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Step 4.2.3.12.1

Factor $3$ out of $3 x$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 1728 - 144 (- \sqrt[3]{3 (x) - 6} + 12) + 578$

Step 4.2.3.12.2

Factor $3$ out of $- 6$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 1728 - 144 (- \sqrt[3]{3 x + 3 \cdot - 2} + 12) + 578$

Step 4.2.3.12.3

Factor $3$ out of $3 x + 3 \cdot - 2$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 1728 - 144 (- \sqrt[3]{3 (x - 2)} + 12) + 578$

Step 4.2.3.13

Apply the distributive property.

Step 4.2.3.14

Multiply $- 1$ by $- 144$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 1728 + 144 \sqrt[3]{3 (x - 2)} - 144 \cdot 12 + 578$

Step 4.2.3.15

Multiply $- 144$ by $12$ .

Step 4.2.4

Simplify by adding terms.

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Step 4.2.4.1

Combine the opposite terms in $x - 578 - 12 \sqrt[3]{9 {(x - 2)}^{2}} + 144 \sqrt[3]{3 (x - 2)} + 12 \sqrt[3]{9 {(x - 2)}^{2}} - 288 \sqrt[3]{3 (x - 2)} + 1728 + 144 \sqrt[3]{3 (x - 2)} - 1728 + 578$ .

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Step 4.2.4.1.1

Add $- 12 \sqrt[3]{9 {(x - 2)}^{2}}$ and $12 \sqrt[3]{9 {(x - 2)}^{2}}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 + 0 + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)} + 1728 + 144 \sqrt[3]{3 (x - 2)} - 1728 + 578$

Step 4.2.4.1.2

Add $x - 578$ and $0$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)} + 1728 + 144 \sqrt[3]{3 (x - 2)} - 1728 + 578$

Step 4.2.4.1.3

Subtract $1728$ from $1728$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)} + 0 + 144 \sqrt[3]{3 (x - 2)} + 578$

Step 4.2.4.1.4

Add $x - 578 + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)}$ and $0$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 578 + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)} + 144 \sqrt[3]{3 (x - 2)} + 578$

Step 4.2.4.1.5

Add $- 578$ and $578$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x + 0 + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)} + 144 \sqrt[3]{3 (x - 2)}$

Step 4.2.4.1.6

Add $x$ and $0$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x + 144 \sqrt[3]{3 (x - 2)} - 288 \sqrt[3]{3 (x - 2)} + 144 \sqrt[3]{3 (x - 2)}$

Step 4.2.4.2

Subtract $288 \sqrt[3]{3 (x - 2)}$ from $144 \sqrt[3]{3 (x - 2)}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x - 144 \sqrt[3]{3 (x - 2)} + 144 \sqrt[3]{3 (x - 2)}$

Step 4.2.4.3

Combine the opposite terms in $x - 144 \sqrt[3]{3 (x - 2)} + 144 \sqrt[3]{3 (x - 2)}$ .

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Step 4.2.4.3.1

Add $- 144 \sqrt[3]{3 (x - 2)}$ and $144 \sqrt[3]{3 (x - 2)}$ .

$f^{- 1} (- \sqrt[3]{3 x - 6} + 12) = x + 0$

Step 4.2.4.3.2

Add $x$ and $0$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) - 6} + 12$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Factor $3$ out of $3 (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) - 6$ .

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Step 4.3.3.1.1

Factor $3$ out of $- 6$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) + 3 \cdot - 2} + 12$

Step 4.3.3.1.2

Factor $3$ out of $3 (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) + 3 \cdot - 2$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578 - 2)} + 12$

Step 4.3.3.2

Subtract $2$ from $578$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 576)} + 12$

Step 4.3.3.3

To write $12 x^{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (- \frac{x^{3}}{3} + 12 x^{2} \cdot \frac{3}{3} - 144 x + 576)} + 12$

Step 4.3.3.4

Combine $12 x^{2}$ and $\frac{3}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (- \frac{x^{3}}{3} + \frac{12 x^{2} \cdot 3}{3} - 144 x + 576)} + 12$

Step 4.3.3.5

Combine the numerators over the common denominator.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 12 x^{2} \cdot 3}{3} - 144 x + 576)} + 12$

Step 4.3.3.6

Simplify the numerator.

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Step 4.3.3.6.1

Factor $x^{2}$ out of $- x^{3} + 12 x^{2} \cdot 3$ .

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Step 4.3.3.6.1.1

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x) + 12 x^{2} \cdot 3}{3} - 144 x + 576)} + 12$

Step 4.3.3.6.1.2

Factor $x^{2}$ out of $12 x^{2} \cdot 3$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x) + x^{2} (12 \cdot 3)}{3} - 144 x + 576)} + 12$

Step 4.3.3.6.1.3

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x + 12 \cdot 3)}{3} - 144 x + 576)} + 12$

Step 4.3.3.6.2

Multiply $12$ by $3$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x + 36)}{3} - 144 x + 576)} + 12$

Step 4.3.3.7

To write $- 144 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x + 36)}{3} - 144 x \cdot \frac{3}{3} + 576)} + 12$

Step 4.3.3.8

Combine $- 144 x$ and $\frac{3}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x + 36)}{3} + \frac{- 144 x \cdot 3}{3} + 576)} + 12$

Step 4.3.3.9

Combine the numerators over the common denominator.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x^{2} (- x + 36) - 144 x \cdot 3}{3} + 576)} + 12$

Step 4.3.3.10

Simplify the numerator.

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Step 4.3.3.10.1

Factor $x$ out of $x^{2} (- x + 36) - 144 x \cdot 3$ .

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Step 4.3.3.10.1.1

Factor $x$ out of $x^{2} (- x + 36)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (x (- x + 36)) - 144 x \cdot 3}{3} + 576)} + 12$

Step 4.3.3.10.1.2

Factor $x$ out of $- 144 x \cdot 3$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (x (- x + 36)) + x (- 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.1.3

Factor $x$ out of $x (x (- x + 36)) + x (- 144 \cdot 3)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (x (- x + 36) - 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.2

Apply the distributive property.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (x (- x) + x \cdot 36 - 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.3

Rewrite using the commutative property of multiplication.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x \cdot x + x \cdot 36 - 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.4

Move $36$ to the left of $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x \cdot x + 36 \cdot x - 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.5

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

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Step 4.3.3.10.5.1

Move $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- (x \cdot x) + 36 \cdot x - 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.5.2

Multiply $x$ by $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2} + 36 \cdot x - 144 \cdot 3)}{3} + 576)} + 12$

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2} + 36 x - 144 \cdot 3)}{3} + 576)} + 12$

Step 4.3.3.10.6

Multiply $- 144$ by $3$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2} + 36 x - 432)}{3} + 576)} + 12$

Step 4.3.3.11

To write $576$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2} + 36 x - 432)}{3} + 576 \cdot \frac{3}{3})} + 12$

Step 4.3.3.12

Combine $576$ and $\frac{3}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2} + 36 x - 432)}{3} + \frac{576 \cdot 3}{3})} + 12$

Step 4.3.3.13

Combine the numerators over the common denominator.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2} + 36 x - 432) + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14

Simplify the numerator.

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Step 4.3.3.14.1

Apply the distributive property.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{x (- x^{2}) + x (36 x) + x \cdot - 432 + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.2

Simplify.

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Step 4.3.3.14.2.1

Rewrite using the commutative property of multiplication.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x \cdot x^{2} + x (36 x) + x \cdot - 432 + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.2.2

Rewrite using the commutative property of multiplication.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x \cdot x^{2} + 36 x \cdot x + x \cdot - 432 + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.2.3

Move $- 432$ to the left of $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x \cdot x^{2} + 36 x \cdot x - 432 \cdot x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.3

Simplify each term.

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Step 4.3.3.14.3.1

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

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Step 4.3.3.14.3.1.1

Move $x^{2}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x^{2} x) + 36 x \cdot x - 432 \cdot x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.3.1.2

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

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Step 4.3.3.14.3.1.2.1

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x^{2} x) + 36 x \cdot x - 432 \cdot x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.3.1.2.2

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{2 + 1} + 36 x \cdot x - 432 \cdot x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.3.1.3

Add $2$ and $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 36 x \cdot x - 432 \cdot x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.3.2

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

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Step 4.3.3.14.3.2.1

Move $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 36 (x \cdot x) - 432 \cdot x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.3.2.2

Multiply $x$ by $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 36 x^{2} - 432 \cdot x + 576 \cdot 3}{3})} + 12$

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 36 x^{2} - 432 x + 576 \cdot 3}{3})} + 12$

Step 4.3.3.14.4

Multiply $576$ by $3$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 36 x^{2} - 432 x + 1728}{3})} + 12$

Step 4.3.3.14.5

Rewrite $- x^{3} + 36 x^{2} - 432 x + 1728$ in a factored form.

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Step 4.3.3.14.5.1

Regroup terms.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- x^{3} + 1728 + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.2

Factor $- 1$ out of $- x^{3} + 1728$ .

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Step 4.3.3.14.5.2.1

Factor $- 1$ out of $- x^{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x^{3}) + 1728 + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.2.2

Rewrite $1728$ as $- 1 (- 1728)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x^{3}) - 1 \cdot - 1728 + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.2.3

Factor $- 1$ out of $- (x^{3}) - 1 (- 1728)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x^{3} - 1728) + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.3

Rewrite $1728$ as $12^{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x^{3} - 12^{3}) + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 12$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- ((x - 12) (x^{2} + x \cdot 12 + 12^{2})) + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.5

Simplify.

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Step 4.3.3.14.5.5.1

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- ((x - 12) (x^{2} + 12 \cdot x + 12^{2})) + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.5.2

Raise $12$ to the power of $2$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- ((x - 12) (x^{2} + 12 x + 144)) + 36 x^{2} - 432 x}{3})} + 12$

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x - 12) (x^{2} + 12 x + 144) + 36 x^{2} - 432 x}{3})} + 12$

Step 4.3.3.14.5.6

Factor $36 x$ out of $36 x^{2} - 432 x$ .

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Step 4.3.3.14.5.6.1

Factor $36 x$ out of $36 x^{2}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x - 12) (x^{2} + 12 x + 144) + 36 x (x) - 432 x}{3})} + 12$

Step 4.3.3.14.5.6.2

Factor $36 x$ out of $- 432 x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x - 12) (x^{2} + 12 x + 144) + 36 x (x) + 36 x (- 12)}{3})} + 12$

Step 4.3.3.14.5.6.3

Factor $36 x$ out of $36 x (x) + 36 x (- 12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- (x - 12) (x^{2} + 12 x + 144) + 36 x (x - 12)}{3})} + 12$

Step 4.3.3.14.5.7

Factor $x - 12$ out of $- (x - 12) (x^{2} + 12 x + 144) + 36 x (x - 12)$ .

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Step 4.3.3.14.5.7.1

Factor $x - 12$ out of $- (x - 12) (x^{2} + 12 x + 144)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- 1 (x^{2} + 12 x + 144)) + 36 x (x - 12)}{3})} + 12$

Step 4.3.3.14.5.7.2

Factor $x - 12$ out of $36 x (x - 12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- 1 (x^{2} + 12 x + 144)) + (x - 12) (36 x)}{3})} + 12$

Step 4.3.3.14.5.7.3

Factor $x - 12$ out of $(x - 12) (- 1 (x^{2} + 12 x + 144)) + (x - 12) (36 x)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- 1 (x^{2} + 12 x + 144) + 36 x)}{3})} + 12$

Step 4.3.3.14.5.8

Apply the distributive property.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- 1 x^{2} - 1 (12 x) - 1 \cdot 144 + 36 x)}{3})} + 12$

Step 4.3.3.14.5.9

Simplify.

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Step 4.3.3.14.5.9.1

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} - 1 (12 x) - 1 \cdot 144 + 36 x)}{3})} + 12$

Step 4.3.3.14.5.9.2

Multiply $12$ by $- 1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} - 12 x - 1 \cdot 144 + 36 x)}{3})} + 12$

Step 4.3.3.14.5.9.3

Multiply $- 1$ by $144$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} - 12 x - 144 + 36 x)}{3})} + 12$

Step 4.3.3.14.5.10

Add $- 12 x$ and $36 x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} + 24 x - 144)}{3})} + 12$

Step 4.3.3.14.5.11

Factor by grouping.

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Step 4.3.3.14.5.11.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 144 = 144$ and whose sum is $b = 24$ .

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Step 4.3.3.14.5.11.1.1

Factor $24$ out of $24 x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} + 24 (x) - 144)}{3})} + 12$

Step 4.3.3.14.5.11.1.2

Rewrite $24$ as $12$ plus $12$

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} + (12 + 12) x - 144)}{3})} + 12$

Step 4.3.3.14.5.11.1.3

Apply the distributive property.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x^{2} + 12 x + 12 x - 144)}{3})} + 12$

Step 4.3.3.14.5.11.2

Factor out the greatest common factor from each group.

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Step 4.3.3.14.5.11.2.1

Group the first two terms and the last two terms.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) ((- x^{2} + 12 x) + 12 x - 144)}{3})} + 12$

Step 4.3.3.14.5.11.2.2

Factor out the greatest common factor (GCF) from each group.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (x (- x + 12) - 12 (- x + 12))}{3})} + 12$

Step 4.3.3.14.5.11.3

Factor the polynomial by factoring out the greatest common factor, $- x + 12$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) ((- x + 12) (x - 12))}{3})} + 12$

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(x - 12) (- x + 12) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12

Combine exponents.

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Step 4.3.3.14.5.12.1

Factor $- 1$ out of $x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(- 1 (- x) - 12) (- x + 12) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.2

Rewrite $- 12$ as $- 1 (12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{(- 1 (- x) - 1 \cdot 12) (- x + 12) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.3

Factor $- 1$ out of $- 1 (- x) - 1 (12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 (- x + 12) (- x + 12) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.4

Raise $- x + 12$ to the power of $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 ((- x + 12) (- x + 12)) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.5

Raise $- x + 12$ to the power of $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 ((- x + 12) (- x + 12)) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.6

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(- x + 12)}^{1 + 1} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.7

Add $1$ and $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(- x + 12)}^{2} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.8

Factor $- 1$ out of $- x$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(- (x) + 12)}^{2} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.9

Rewrite $12$ as $- 1 (- 12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(- (x) - 1 \cdot - 12)}^{2} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.10

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(- (x - 12))}^{2} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.11

Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(- 1 (x - 12))}^{2} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.12

Apply the product rule to $- 1 (x - 12)$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 ({(- 1)}^{2} {(x - 12)}^{2}) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.13

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 (1 {(x - 12)}^{2}) (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.14

Multiply ${(x - 12)}^{2}$ by $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(x - 12)}^{2} (x - 12)}{3})} + 12$

Step 4.3.3.14.5.12.15

Raise $x - 12$ to the power of $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 ((x - 12) {(x - 12)}^{2})}{3})} + 12$

Step 4.3.3.14.5.12.16

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(x - 12)}^{1 + 2}}{3})} + 12$

Step 4.3.3.14.5.12.17

Add $1$ and $2$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 (\frac{- 1 {(x - 12)}^{3}}{3})} + 12$

Step 4.3.3.15

Move the negative in front of the fraction.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{3 \cdot (- 1 \frac{{(x - 12)}^{3}}{3})} + 12$

Step 4.3.3.16

Combine exponents.

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Step 4.3.3.16.1

Factor out negative.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{- (3 (\frac{{(x - 12)}^{3}}{3}))} + 12$

Step 4.3.3.16.2

Combine $3$ and $\frac{{(x - 12)}^{3}}{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{- \frac{3 {(x - 12)}^{3}}{3}} + 12$

Step 4.3.3.17

Reduce the expression by cancelling the common factors.

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Step 4.3.3.17.1

Reduce the expression $\frac{3 {(x - 12)}^{3}}{3}$ by cancelling the common factors.

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Step 4.3.3.17.1.1

Cancel the common factor.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{- \frac{3 {(x - 12)}^{3}}{3}} + 12$

Step 4.3.3.17.1.2

Rewrite the expression.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{- \frac{{(x - 12)}^{3}}{1}} + 12$

Step 4.3.3.17.2

Divide ${(x - 12)}^{3}$ by $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{- {(x - 12)}^{3}} + 12$

Step 4.3.3.18

Rewrite $- {(x - 12)}^{3}$ as ${(- (x - 12))}^{3}$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - \sqrt[3]{{(- (x - 12))}^{3}} + 12$

Step 4.3.3.19

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

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = (x - 12) + 12$

Step 4.3.3.20

Apply the distributive property.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - (- x + 12) + 12$

Step 4.3.3.21

Multiply $- 1$ by $- 12$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = - (- x + 12) + 12$

Step 4.3.3.22

Apply the distributive property.

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = x - 1 \cdot 12 + 12$

Step 4.3.3.23

Multiply $- - x$ .

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Step 4.3.3.23.1

Multiply $- 1$ by $- 1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = 1 x - 1 \cdot 12 + 12$

Step 4.3.3.23.2

Multiply $x$ by $1$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = x - 1 \cdot 12 + 12$

Step 4.3.3.24

Multiply $- 1$ by $12$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = x - 12 + 12$

Step 4.3.4

Combine the opposite terms in $x - 12 + 12$ .

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Step 4.3.4.1

Add $- 12$ and $12$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

$f (- \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578$ is the inverse of $f (x) = - \sqrt[3]{3 x - 6} + 12$ .

$f^{- 1} (x) = - \frac{x^{3}}{3} + 12 x^{2} - 144 x + 578$