Algebra Examples

$f (x) = 3 (\sqrt[3]{x} - 1) + 10$

Step 1

Write $f (x) = 3 (\sqrt[3]{x} - 1) + 10$ as an equation.

$y = 3 (\sqrt[3]{x} - 1) + 10$

Step 2

Interchange the variables.

$x = 3 (\sqrt[3]{y} - 1) + 10$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $3 (\sqrt[3]{y} - 1) + 10 = x$ .

$3 (\sqrt[3]{y} - 1) + 10 = x$

Step 3.2

Solve for $\sqrt[3]{y}$ .

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Step 3.2.1

Simplify $3 (\sqrt[3]{y} - 1) + 10$ .

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Step 3.2.1.1

Simplify each term.

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Step 3.2.1.1.1

Apply the distributive property.

$3 \sqrt[3]{y} + 3 \cdot - 1 + 10 = x$

Step 3.2.1.1.2

Multiply $3$ by $- 1$ .

$3 \sqrt[3]{y} - 3 + 10 = x$

Step 3.2.1.2

Add $- 3$ and $10$ .

$3 \sqrt[3]{y} + 7 = x$

Step 3.2.2

Subtract $7$ from both sides of the equation.

$3 \sqrt[3]{y} = x - 7$

Step 3.2.3

Divide each term in $3 \sqrt[3]{y} = x - 7$ by $3$ and simplify.

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Step 3.2.3.1

Divide each term in $3 \sqrt[3]{y} = x - 7$ by $3$ .

$\frac{3 \sqrt[3]{y}}{3} = \frac{x}{3} + \frac{- 7}{3}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $3$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{3 \sqrt[3]{y}}{3} = \frac{x}{3} + \frac{- 7}{3}$

Step 3.2.3.2.1.2

Divide $\sqrt[3]{y}$ by $1$ .

$\sqrt[3]{y} = \frac{x}{3} + \frac{- 7}{3}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Move the negative in front of the fraction.

$\sqrt[3]{y} = \frac{x}{3} - \frac{7}{3}$

Step 3.3

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

${\sqrt[3]{y}}^{3} = {(\frac{x}{3} - \frac{7}{3})}^{3}$

Step 3.4

Simplify each side of the equation.

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Step 3.4.1

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

${(y^{\frac{1}{3}})}^{3} = {(\frac{x}{3} - \frac{7}{3})}^{3}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

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

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Step 3.4.2.1.1

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

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Step 3.4.2.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{3} - \frac{7}{3})}^{3}$

Step 3.4.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(\frac{x}{3} - \frac{7}{3})}^{3}$

Step 3.4.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{x}{3} - \frac{7}{3})}^{3}$

Step 3.4.2.1.2

Simplify.

$y = {(\frac{x}{3} - \frac{7}{3})}^{3}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Simplify ${(\frac{x}{3} - \frac{7}{3})}^{3}$ .

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Step 3.4.3.1.1

Use the Binomial Theorem.

$y = {(\frac{x}{3})}^{3} + 3 {(\frac{x}{3})}^{2} (- \frac{7}{3}) + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Apply the product rule to $\frac{x}{3}$ .

$y = \frac{x^{3}}{3^{3}} + 3 {(\frac{x}{3})}^{2} (- \frac{7}{3}) + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.2

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

$y = \frac{x^{3}}{27} + 3 {(\frac{x}{3})}^{2} (- \frac{7}{3}) + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.3

Cancel the common factor of $3$ .

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Step 3.4.3.1.2.3.1

Move the leading negative in $- \frac{7}{3}$ into the numerator.

$y = \frac{x^{3}}{27} + 3 {(\frac{x}{3})}^{2} \frac{- 7}{3} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.3.2

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

$y = \frac{x^{3}}{27} + 3 ({(\frac{x}{3})}^{2}) \frac{- 7}{3} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.3.3

Cancel the common factor.

$y = \frac{x^{3}}{27} + 3 {(\frac{x}{3})}^{2} \frac{- 7}{3} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.3.4

Rewrite the expression.

$y = \frac{x^{3}}{27} + {(\frac{x}{3})}^{2} \cdot - 7 + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.4

Apply the product rule to $\frac{x}{3}$ .

$y = \frac{x^{3}}{27} + \frac{x^{2}}{3^{2}} \cdot - 7 + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.5

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

$y = \frac{x^{3}}{27} + \frac{x^{2}}{9} \cdot - 7 + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.6

Combine $\frac{x^{2}}{9}$ and $- 7$ .

$y = \frac{x^{3}}{27} + \frac{x^{2} \cdot - 7}{9} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.7

Move $- 7$ to the left of $x^{2}$ .

$y = \frac{x^{3}}{27} + \frac{- 7 \cdot x^{2}}{9} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.8

Move the negative in front of the fraction.

$y = \frac{x^{3}}{27} - \frac{7 \cdot x^{2}}{9} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.9

Cancel the common factor of $3$ .

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Step 3.4.3.1.2.9.1

Cancel the common factor.

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + 3 \frac{x}{3} {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.9.2

Rewrite the expression.

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x {(- \frac{7}{3})}^{2} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.10

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.3.1.2.10.1

Apply the product rule to $- \frac{7}{3}$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x ({(- 1)}^{2} {(\frac{7}{3})}^{2}) + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.10.2

Apply the product rule to $\frac{7}{3}$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x ({(- 1)}^{2} \frac{7^{2}}{3^{2}}) + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.11

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

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x (1 \frac{7^{2}}{3^{2}}) + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.12

Multiply $\frac{7^{2}}{3^{2}}$ by $1$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x \frac{7^{2}}{3^{2}} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.13

Raise $7$ to the power of $2$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x \frac{49}{3^{2}} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.14

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

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + x \frac{49}{9} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.15

Combine $x$ and $\frac{49}{9}$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{x \cdot 49}{9} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.16

Move $49$ to the left of $x$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 \cdot x}{9} + {(- \frac{7}{3})}^{3}$

Step 3.4.3.1.2.17

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.4.3.1.2.17.1

Apply the product rule to $- \frac{7}{3}$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} + {(- 1)}^{3} {(\frac{7}{3})}^{3}$

Step 3.4.3.1.2.17.2

Apply the product rule to $\frac{7}{3}$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} + {(- 1)}^{3} \frac{7^{3}}{3^{3}}$

Step 3.4.3.1.2.18

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

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{7^{3}}{3^{3}}$

Step 3.4.3.1.2.19

Raise $7$ to the power of $3$ .

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{3^{3}}$

Step 3.4.3.1.2.20

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

$y = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}$

Step 4

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

$f^{- 1} (x) = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}$

Step 5

Verify if $f^{- 1} (x) = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}$ is the inverse of $f (x) = 3 (\sqrt[3]{x} - 1) + 10$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (3 (\sqrt[3]{x} - 1) + 10)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} - \frac{7 {(3 (\sqrt[3]{x} - 1) + 10)}^{2}}{9} + \frac{49 (3 (\sqrt[3]{x} - 1) + 10)}{9} - \frac{343}{27}$

Step 5.2.3

Simplify terms.

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Step 5.2.3.1

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 {(3 (\sqrt[3]{x} - 1) + 10)}^{2} + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2

Simplify each term.

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Step 5.2.3.2.1

Simplify each term.

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Step 5.2.3.2.1.1

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 {(3 \sqrt[3]{x} + 3 \cdot - 1 + 10)}^{2} + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.1.2

Multiply $3$ by $- 1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 {(3 \sqrt[3]{x} - 3 + 10)}^{2} + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.2

Add $- 3$ and $10$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 {(3 \sqrt[3]{x} + 7)}^{2} + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.3

Rewrite ${(3 \sqrt[3]{x} + 7)}^{2}$ as $(3 \sqrt[3]{x} + 7) (3 \sqrt[3]{x} + 7)$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 ((3 \sqrt[3]{x} + 7) (3 \sqrt[3]{x} + 7)) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.4

Expand $(3 \sqrt[3]{x} + 7) (3 \sqrt[3]{x} + 7)$ using the FOIL Method.

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Step 5.2.3.2.4.1

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (3 \sqrt[3]{x} (3 \sqrt[3]{x} + 7) + 7 (3 \sqrt[3]{x} + 7)) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.4.2

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (3 \sqrt[3]{x} (3 \sqrt[3]{x}) + 3 \sqrt[3]{x} \cdot 7 + 7 (3 \sqrt[3]{x} + 7)) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.4.3

Apply the distributive property.

Step 5.2.3.2.5

Simplify and combine like terms.

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Step 5.2.3.2.5.1

Simplify each term.

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Step 5.2.3.2.5.1.1

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

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Step 5.2.3.2.5.1.1.1

Multiply $3$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x} \sqrt[3]{x} + 3 \sqrt[3]{x} \cdot 7 + 7 (3 \sqrt[3]{x}) + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.1.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 (\sqrt[3]{x} \sqrt[3]{x}) + 3 \sqrt[3]{x} \cdot 7 + 7 (3 \sqrt[3]{x}) + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.1.3

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

Step 5.2.3.2.5.1.1.4

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 {\sqrt[3]{x}}^{1 + 1} + 3 \sqrt[3]{x} \cdot 7 + 7 (3 \sqrt[3]{x}) + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.1.5

Add $1$ and $1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 {\sqrt[3]{x}}^{2} + 3 \sqrt[3]{x} \cdot 7 + 7 (3 \sqrt[3]{x}) + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 7 + 7 (3 \sqrt[3]{x}) + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.3

Multiply $7$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x^{2}} + 21 \sqrt[3]{x} + 7 (3 \sqrt[3]{x}) + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.4

Multiply $3$ by $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x^{2}} + 21 \sqrt[3]{x} + 21 \sqrt[3]{x} + 7 \cdot 7) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.1.5

Multiply $7$ by $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x^{2}} + 21 \sqrt[3]{x} + 21 \sqrt[3]{x} + 49) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.5.2

Add $21 \sqrt[3]{x}$ and $21 \sqrt[3]{x}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x^{2}} + 42 \sqrt[3]{x} + 49) + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.6

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 7 (9 \sqrt[3]{x^{2}}) - 7 (42 \sqrt[3]{x}) - 7 \cdot 49 + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.7

Simplify.

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Step 5.2.3.2.7.1

Multiply $9$ by $- 7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 7 (42 \sqrt[3]{x}) - 7 \cdot 49 + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.7.2

Multiply $42$ by $- 7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 7 \cdot 49 + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.7.3

Multiply $- 7$ by $49$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 49 (3 (\sqrt[3]{x} - 1) + 10)}{9}$

Step 5.2.3.2.8

Simplify each term.

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Step 5.2.3.2.8.1

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 49 (3 \sqrt[3]{x} + 3 \cdot - 1 + 10)}{9}$

Step 5.2.3.2.8.2

Multiply $3$ by $- 1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 49 (3 \sqrt[3]{x} - 3 + 10)}{9}$

Step 5.2.3.2.9

Add $- 3$ and $10$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 49 (3 \sqrt[3]{x} + 7)}{9}$

Step 5.2.3.2.10

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 49 (3 \sqrt[3]{x}) + 49 \cdot 7}{9}$

Step 5.2.3.2.11

Multiply $3$ by $49$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 147 \sqrt[3]{x} + 49 \cdot 7}{9}$

Step 5.2.3.2.12

Multiply $49$ by $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 147 \sqrt[3]{x} + 343}{9}$

Step 5.2.3.3

Simplify by adding terms.

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Step 5.2.3.3.1

Combine the opposite terms in $- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} - 343 + 147 \sqrt[3]{x} + 343$ .

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Step 5.2.3.3.1.1

Add $- 343$ and $343$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 0 + 147 \sqrt[3]{x}}{9}$

Step 5.2.3.3.1.2

Add $- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 294 \sqrt[3]{x} + 147 \sqrt[3]{x}}{9}$

Step 5.2.3.3.2

Add $- 294 \sqrt[3]{x}$ and $147 \sqrt[3]{x}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 (\sqrt[3]{x} - 1) + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 147 \sqrt[3]{x}}{9}$

Step 5.2.3.4

Simplify each term.

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Step 5.2.3.4.1

Simplify the numerator.

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Step 5.2.3.4.1.1

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 3 \cdot - 1 + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 147 \sqrt[3]{x}}{9}$

Step 5.2.3.4.1.2

Multiply $3$ by $- 1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} - 3 + 10)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 147 \sqrt[3]{x}}{9}$

Step 5.2.3.4.1.3

Add $- 3$ and $10$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} + \frac{- 343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 147 \sqrt[3]{x}}{9}$

Step 5.2.3.4.2

Move the negative in front of the fraction.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 63 \sqrt[3]{x^{2}} - 147 \sqrt[3]{x}}{9}$

Step 5.2.3.4.3

Cancel the common factor of $- 63 \sqrt[3]{x^{2}} - 147 \sqrt[3]{x}$ and $9$ .

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Step 5.2.3.4.3.1

Factor $3$ out of $- 63 \sqrt[3]{x^{2}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{3 (- 21 \sqrt[3]{x^{2}}) - 147 \sqrt[3]{x}}{9}$

Step 5.2.3.4.3.2

Factor $3$ out of $- 147 \sqrt[3]{x}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{3 (- 21 \sqrt[3]{x^{2}}) + 3 (- 49 \sqrt[3]{x})}{9}$

Step 5.2.3.4.3.3

Factor $3$ out of $3 (- 21 \sqrt[3]{x^{2}}) + 3 (- 49 \sqrt[3]{x})$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{3 (- 21 \sqrt[3]{x^{2}} - 49 \sqrt[3]{x})}{9}$

Step 5.2.3.4.3.4

Cancel the common factors.

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Step 5.2.3.4.3.4.1

Factor $3$ out of $9$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{3 (- 21 \sqrt[3]{x^{2}} - 49 \sqrt[3]{x})}{3 (3)}$

Step 5.2.3.4.3.4.2

Cancel the common factor.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{3 (- 21 \sqrt[3]{x^{2}} - 49 \sqrt[3]{x})}{3 \cdot 3}$

Step 5.2.3.4.3.4.3

Rewrite the expression.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 21 \sqrt[3]{x^{2}} - 49 \sqrt[3]{x}}{3}$

Step 5.2.3.4.4

Simplify the numerator.

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Step 5.2.3.4.4.1

Factor $- 7$ out of $- 21 \sqrt[3]{x^{2}} - 49 \sqrt[3]{x}$ .

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Step 5.2.3.4.4.1.1

Factor $- 7$ out of $- 21 \sqrt[3]{x^{2}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (3 \sqrt[3]{x^{2}}) - 49 \sqrt[3]{x}}{3}$

Step 5.2.3.4.4.1.2

Factor $- 7$ out of $- 49 \sqrt[3]{x}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (3 \sqrt[3]{x^{2}}) - 7 (7 \sqrt[3]{x})}{3}$

Step 5.2.3.4.4.1.3

Factor $- 7$ out of $- 7 (3 \sqrt[3]{x^{2}}) - 7 (7 \sqrt[3]{x})$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (3 \sqrt[3]{x^{2}} + 7 \sqrt[3]{x})}{3}$

Step 5.2.3.4.4.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (3 x^{\frac{2}{3}} + 7 \sqrt[3]{x})}{3}$

Step 5.2.3.4.4.3

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (3 x^{\frac{2}{3}} + 7 x^{\frac{1}{3}})}{3}$

Step 5.2.3.4.4.4

Factor $x^{\frac{1}{3}}$ out of $3 x^{\frac{2}{3}} + 7 x^{\frac{1}{3}}$ .

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Step 5.2.3.4.4.4.1

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 7 x^{\frac{1}{3}})}{3}$

Step 5.2.3.4.4.4.2

Factor $x^{\frac{1}{3}}$ out of $7 x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + x^{\frac{1}{3}} \cdot 7)}{3}$

Step 5.2.3.4.4.4.3

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + x^{\frac{1}{3}} \cdot 7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 (x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7))}{3}$

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} + \frac{- 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.4.5

Move the negative in front of the fraction.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3}}{27} - \frac{343}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.5

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3} - 343}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6

Simplify each term.

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Step 5.2.3.6.1

Simplify the numerator.

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Step 5.2.3.6.1.1

Rewrite $343$ as $7^{3}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{{(3 \sqrt[3]{x} + 7)}^{3} - 7^{3}}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.2

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 $a = 3 \sqrt[3]{x} + 7$ and $b = 7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{(3 \sqrt[3]{x} + 7 - 7) ({(3 \sqrt[3]{x} + 7)}^{2} + (3 \sqrt[3]{x} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3

Simplify.

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Step 5.2.3.6.1.3.1

Subtract $7$ from $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{(3 \sqrt[3]{x} + 0) ({(3 \sqrt[3]{x} + 7)}^{2} + (3 \sqrt[3]{x} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} ({(3 \sqrt[3]{x} + 7)}^{2} + (3 \sqrt[3]{x} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.3

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} ({(3 x^{\frac{1}{3}} + 7)}^{2} + (3 \sqrt[3]{x} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.4

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} ({(3 x^{\frac{1}{3}} + 7)}^{2} + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.5

Rewrite ${(3 x^{\frac{1}{3}} + 7)}^{2}$ as $(3 x^{\frac{1}{3}} + 7) (3 x^{\frac{1}{3}} + 7)$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} ((3 x^{\frac{1}{3}} + 7) (3 x^{\frac{1}{3}} + 7) + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.6

Expand $(3 x^{\frac{1}{3}} + 7) (3 x^{\frac{1}{3}} + 7)$ using the FOIL Method.

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Step 5.2.3.6.1.3.6.1

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7) + 7 (3 x^{\frac{1}{3}} + 7) + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.6.2

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}} + 7) + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.6.3

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7

Simplify and combine like terms.

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Step 5.2.3.6.1.3.7.1

Simplify each term.

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Step 5.2.3.6.1.3.7.1.1

Rewrite using the commutative property of multiplication.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 \cdot (3 x^{\frac{1}{3}} x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.2

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

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Step 5.2.3.6.1.3.7.1.2.1

Move $x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 \cdot (3 (x^{\frac{1}{3}} x^{\frac{1}{3}})) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.2.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 \cdot (3 x^{\frac{1}{3} + \frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.2.3

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 \cdot (3 x^{\frac{1 + 1}{3}}) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.2.4

Add $1$ and $1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 \cdot (3 x^{\frac{2}{3}}) + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.3

Multiply $3$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot 7 + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.4

Multiply $7$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 21 x^{\frac{1}{3}} + 7 (3 x^{\frac{1}{3}}) + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.5

Multiply $3$ by $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 21 x^{\frac{1}{3}} + 21 x^{\frac{1}{3}} + 7 \cdot 7 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.1.6

Multiply $7$ by $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 21 x^{\frac{1}{3}} + 21 x^{\frac{1}{3}} + 49 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.7.2

Add $21 x^{\frac{1}{3}}$ and $21 x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 42 x^{\frac{1}{3}} + 49 + (3 x^{\frac{1}{3}} + 7) \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.8

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 42 x^{\frac{1}{3}} + 49 + 3 x^{\frac{1}{3}} \cdot 7 + 7 \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.9

Multiply $7$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 42 x^{\frac{1}{3}} + 49 + 21 x^{\frac{1}{3}} + 7 \cdot 7 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.10

Multiply $7$ by $7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 42 x^{\frac{1}{3}} + 49 + 21 x^{\frac{1}{3}} + 49 + 7^{2})}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.11

Raise $7$ to the power of $2$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 42 x^{\frac{1}{3}} + 49 + 21 x^{\frac{1}{3}} + 49 + 49)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.12

Add $42 x^{\frac{1}{3}}$ and $21 x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 63 x^{\frac{1}{3}} + 49 + 49 + 49)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.13

Add $49$ and $49$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 63 x^{\frac{1}{3}} + 98 + 49)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.14

Add $98$ and $49$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (9 x^{\frac{2}{3}} + 63 x^{\frac{1}{3}} + 147)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.15

Reorder terms.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (63 x^{\frac{1}{3}} + 9 x^{\frac{2}{3}} + 147)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.16

Factor $3$ out of $63 x^{\frac{1}{3}} + 9 x^{\frac{2}{3}} + 147$ .

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Step 5.2.3.6.1.3.16.1

Factor $3$ out of $63 x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 (21 x^{\frac{1}{3}}) + 9 x^{\frac{2}{3}} + 147)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.16.2

Factor $3$ out of $9 x^{\frac{2}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 (21 x^{\frac{1}{3}}) + 3 (3 x^{\frac{2}{3}}) + 147)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.16.3

Factor $3$ out of $147$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 (21 x^{\frac{1}{3}}) + 3 (3 x^{\frac{2}{3}}) + 3 (49))}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.16.4

Factor $3$ out of $3 (21 x^{\frac{1}{3}}) + 3 (3 x^{\frac{2}{3}})$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}}) + 3 (49))}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.16.5

Factor $3$ out of $3 (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}}) + 3 (49)$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} (3 (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49))}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 \sqrt[3]{x} \cdot (3 (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49))}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.1.3.17

Multiply $3$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{9 \sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49)}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.2

Factor $9$ out of $9 \sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49)$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{9 (\sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49))}{27} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.3

Cancel the common factors.

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Step 5.2.3.6.3.1

Factor $9$ out of $27$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{9 (\sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49))}{9 \cdot 3} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.3.2

Cancel the common factor.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{9 (\sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49))}{9 \cdot 3} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.6.3.3

Rewrite the expression.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{\sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49)}{3} - \frac{7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.7

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{\sqrt[3]{x} (21 x^{\frac{1}{3}} + 3 x^{\frac{2}{3}} + 49) - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8

Simplify each term.

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Step 5.2.3.8.1

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{\sqrt[3]{x} (21 x^{\frac{1}{3}}) + \sqrt[3]{x} (3 x^{\frac{2}{3}}) + \sqrt[3]{x} \cdot 49 - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.2

Simplify.

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Step 5.2.3.8.2.1

Rewrite using the commutative property of multiplication.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 \sqrt[3]{x} x^{\frac{1}{3}} + \sqrt[3]{x} (3 x^{\frac{2}{3}}) + \sqrt[3]{x} \cdot 49 - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.2.2

Rewrite using the commutative property of multiplication.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 \sqrt[3]{x} x^{\frac{1}{3}} + 3 \sqrt[3]{x} x^{\frac{2}{3}} + \sqrt[3]{x} \cdot 49 - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.2.3

Move $49$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 \sqrt[3]{x} x^{\frac{1}{3}} + 3 \sqrt[3]{x} x^{\frac{2}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3

Simplify each term.

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Step 5.2.3.8.3.1

Multiply $21 \sqrt[3]{x} x^{\frac{1}{3}}$ .

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Step 5.2.3.8.3.1.1

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 (x^{\frac{1}{3}} x^{\frac{1}{3}}) + 3 \sqrt[3]{x} x^{\frac{2}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.1.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{1}{3} + \frac{1}{3}} + 3 \sqrt[3]{x} x^{\frac{2}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.1.3

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{1 + 1}{3}} + 3 \sqrt[3]{x} x^{\frac{2}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.1.4

Add $1$ and $1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 \sqrt[3]{x} x^{\frac{2}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.2

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

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Step 5.2.3.8.3.2.1

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 (x^{\frac{2}{3}} x^{\frac{1}{3}}) + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.2.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x^{\frac{2}{3} + \frac{1}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.2.3

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x^{\frac{2 + 1}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.2.4

Add $2$ and $1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x^{\frac{3}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.2.5

Cancel the common factor of $3$ .

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Step 5.2.3.8.3.2.5.1

Cancel the common factor.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x^{\frac{3}{3}} + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.3.2.5.2

Rewrite the expression.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \cdot \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} + 7)}{3}$

Step 5.2.3.8.4

Apply the distributive property.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) - 7 x^{\frac{1}{3}} \cdot 7}{3}$

Step 5.2.3.8.5

Rewrite using the commutative property of multiplication.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 \cdot (3 x^{\frac{1}{3}} x^{\frac{1}{3}}) - 7 x^{\frac{1}{3}} \cdot 7}{3}$

Step 5.2.3.8.6

Multiply $7$ by $- 7$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 \cdot (3 x^{\frac{1}{3}} x^{\frac{1}{3}}) - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.8.7

Simplify each term.

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Step 5.2.3.8.7.1

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

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Step 5.2.3.8.7.1.1

Move $x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 \cdot (3 (x^{\frac{1}{3}} x^{\frac{1}{3}})) - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.8.7.1.2

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 \cdot (3 x^{\frac{1}{3} + \frac{1}{3}}) - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.8.7.1.3

Combine the numerators over the common denominator.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 \cdot (3 x^{\frac{1 + 1}{3}}) - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.8.7.1.4

Add $1$ and $1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 7 \cdot (3 x^{\frac{2}{3}}) - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.8.7.2

Multiply $- 7$ by $3$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 21 x^{\frac{2}{3}} - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.9

Combine the opposite terms in $21 x^{\frac{2}{3}} + 3 x + 49 \sqrt[3]{x} - 21 x^{\frac{2}{3}} - 49 x^{\frac{1}{3}}$ .

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Step 5.2.3.9.1

Subtract $21 x^{\frac{2}{3}}$ from $21 x^{\frac{2}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{0 + 3 x + 49 \sqrt[3]{x} - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.3.9.2

Add $0$ and $3 x$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 x + 49 \sqrt[3]{x} - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.4

Simplify the numerator.

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Step 5.2.4.1

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

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 x + 49 x^{\frac{1}{3}} - 49 x^{\frac{1}{3}}}{3}$

Step 5.2.4.2

Subtract $49 x^{\frac{1}{3}}$ from $49 x^{\frac{1}{3}}$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 x + 0}{3}$

Step 5.2.4.3

Add $3 x$ and $0$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 x}{3}$

Step 5.2.5

Cancel the common factor of $3$ .

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Step 5.2.5.1

Cancel the common factor.

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = \frac{3 x}{3}$

Step 5.2.5.2

Divide $x$ by $1$ .

$f^{- 1} (3 (\sqrt[3]{x} - 1) + 10) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.3

Remove parentheses.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Simplify each term.

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Step 5.3.4.1.1

To write $- \frac{7 x^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3}}{27} - \frac{7 x^{2}}{9} \cdot \frac{3}{3} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.2

Write each expression with a common denominator of $27$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.4.1.2.1

Multiply $\frac{7 x^{2}}{9}$ by $\frac{3}{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3}}{27} - \frac{7 x^{2} \cdot 3}{9 \cdot 3} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.2.2

Multiply $9$ by $3$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3}}{27} - \frac{7 x^{2} \cdot 3}{27} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.3

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3} - 7 x^{2} \cdot 3}{27} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.4

Simplify the numerator.

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Step 5.3.4.1.4.1

Factor $x^{2}$ out of $x^{3} - 7 x^{2} \cdot 3$ .

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Step 5.3.4.1.4.1.1

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} x - 7 x^{2} \cdot 3}{27} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.4.1.2

Factor $x^{2}$ out of $- 7 x^{2} \cdot 3$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} x + x^{2} (- 7 \cdot 3)}{27} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.4.1.3

Factor $x^{2}$ out of $x^{2} x + x^{2} (- 7 \cdot 3)$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 7 \cdot 3)}{27} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.4.2

Multiply $- 7$ by $3$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21)}{27} + \frac{49 x}{9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.5

Find the common denominator.

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Step 5.3.4.1.5.1

Multiply $\frac{49 x}{9}$ by $\frac{3}{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21)}{27} + \frac{49 x}{9} \cdot \frac{3}{3} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.5.2

Multiply $\frac{49 x}{9}$ by $\frac{3}{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21)}{27} + \frac{49 x \cdot 3}{9 \cdot 3} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.5.3

Reorder the factors of $9 \cdot 3$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21)}{27} + \frac{49 x \cdot 3}{3 \cdot 9} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.5.4

Multiply $3$ by $9$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21)}{27} + \frac{49 x \cdot 3}{27} - \frac{343}{27}} - 1) + 10$

Step 5.3.4.1.6

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21) + 49 x \cdot 3 - 343}{27}} - 1) + 10$

Step 5.3.4.1.7

Multiply $3$ by $49$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} (x - 21) + 147 x - 343}{27}} - 1) + 10$

Step 5.3.4.1.8

Simplify the numerator.

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Step 5.3.4.1.8.1

Apply the distributive property.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} x + x^{2} \cdot - 21 + 147 x - 343}{27}} - 1) + 10$

Step 5.3.4.1.8.2

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

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Step 5.3.4.1.8.2.1

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

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Step 5.3.4.1.8.2.1.1

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2} x + x^{2} \cdot - 21 + 147 x - 343}{27}} - 1) + 10$

Step 5.3.4.1.8.2.1.2

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{2 + 1} + x^{2} \cdot - 21 + 147 x - 343}{27}} - 1) + 10$

Step 5.3.4.1.8.2.2

Add $2$ and $1$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3} + x^{2} \cdot - 21 + 147 x - 343}{27}} - 1) + 10$

Step 5.3.4.1.8.3

Move $- 21$ to the left of $x^{2}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{x^{3} - 21 \cdot x^{2} + 147 x - 343}{27}} - 1) + 10$

Step 5.3.4.1.8.4

Rewrite $x^{3} - 21 x^{2} + 147 x - 343$ in a factored form.

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Step 5.3.4.1.8.4.1

Factor $x^{3} - 21 x^{2} + 147 x - 343$ using the rational roots test.

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Step 5.3.4.1.8.4.1.1

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.

$p = \pm 1, \pm 343, \pm 7, \pm 49$

$q = \pm 1$

Step 5.3.4.1.8.4.1.2

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

$\pm 1, \pm 343, \pm 7, \pm 49$

Step 5.3.4.1.8.4.1.3

Substitute $7$ and simplify the expression. In this case, the expression is equal to $0$ so $7$ is a root of the polynomial.

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Step 5.3.4.1.8.4.1.3.1

Substitute $7$ into the polynomial.

$7^{3} - 21 \cdot 7^{2} + 147 \cdot 7 - 343$

Step 5.3.4.1.8.4.1.3.2

Raise $7$ to the power of $3$ .

$343 - 21 \cdot 7^{2} + 147 \cdot 7 - 343$

Step 5.3.4.1.8.4.1.3.3

Raise $7$ to the power of $2$ .

$343 - 21 \cdot 49 + 147 \cdot 7 - 343$

Step 5.3.4.1.8.4.1.3.4

Multiply $- 21$ by $49$ .

$343 - 1029 + 147 \cdot 7 - 343$

Step 5.3.4.1.8.4.1.3.5

Subtract $1029$ from $343$ .

$- 686 + 147 \cdot 7 - 343$

Step 5.3.4.1.8.4.1.3.6

Multiply $147$ by $7$ .

$- 686 + 1029 - 343$

Step 5.3.4.1.8.4.1.3.7

Add $- 686$ and $1029$ .

$343 - 343$

Step 5.3.4.1.8.4.1.3.8

Subtract $343$ from $343$ .

$0$

Step 5.3.4.1.8.4.1.4

Since $7$ is a known root, divide the polynomial by $x - 7$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 21 x^{2} + 147 x - 343}{x - 7}$

Step 5.3.4.1.8.4.1.5

Divide $x^{3} - 21 x^{2} + 147 x - 343$ by $x - 7$ .

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Step 5.3.4.1.8.4.1.5.1

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

$x$

$7$

$x^{3}$

$21 x^{2}$

$147 x$

$343$

Step 5.3.4.1.8.4.1.5.2

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

					$x^{2}$
	$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$

Step 5.3.4.1.8.4.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			+	$x^{3}$	-	$7 x^{2}$

Step 5.3.4.1.8.4.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 7 x^{2}$

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$

Step 5.3.4.1.8.4.1.5.5

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

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$

Step 5.3.4.1.8.4.1.5.6

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

				$x^{2}$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$

Step 5.3.4.1.8.4.1.5.7

Divide the highest order term in the dividend $- 14 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$14 x$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$

Step 5.3.4.1.8.4.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$14 x$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					-	$14 x^{2}$	+	$98 x$

Step 5.3.4.1.8.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{2} + 98 x$

				$x^{2}$	-	$14 x$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$

Step 5.3.4.1.8.4.1.5.10

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

				$x^{2}$	-	$14 x$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$
							+	$49 x$

Step 5.3.4.1.8.4.1.5.11

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

				$x^{2}$	-	$14 x$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$
							+	$49 x$	-	$343$

Step 5.3.4.1.8.4.1.5.12

Divide the highest order term in the dividend $49 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$14 x$	+	$49$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$
							+	$49 x$	-	$343$

Step 5.3.4.1.8.4.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$14 x$	+	$49$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$
							+	$49 x$	-	$343$
							+	$49 x$	-	$343$

Step 5.3.4.1.8.4.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $49 x - 343$

				$x^{2}$	-	$14 x$	+	$49$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$
							+	$49 x$	-	$343$
							-	$49 x$	+	$343$

Step 5.3.4.1.8.4.1.5.15

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

				$x^{2}$	-	$14 x$	+	$49$
$x$	-	$7$		$x^{3}$	-	$21 x^{2}$	+	$147 x$	-	$343$
			-	$x^{3}$	+	$7 x^{2}$
					-	$14 x^{2}$	+	$147 x$
					+	$14 x^{2}$	-	$98 x$
							+	$49 x$	-	$343$
							-	$49 x$	+	$343$
										$0$

Step 5.3.4.1.8.4.1.5.16

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

$x^{2} - 14 x + 49$

Step 5.3.4.1.8.4.1.6

Write $x^{3} - 21 x^{2} + 147 x - 343$ as a set of factors.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{(x - 7) (x^{2} - 14 x + 49)}{27}} - 1) + 10$

Step 5.3.4.1.8.4.2

Factor using the perfect square rule.

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Step 5.3.4.1.8.4.2.1

Rewrite $49$ as $7^{2}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{(x - 7) (x^{2} - 14 x + 7^{2})}{27}} - 1) + 10$

Step 5.3.4.1.8.4.2.2

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

$14 x = 2 \cdot x \cdot 7$

Step 5.3.4.1.8.4.2.3

Rewrite the polynomial.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{(x - 7) (x^{2} - 2 \cdot x \cdot 7 + 7^{2})}{27}} - 1) + 10$

Step 5.3.4.1.8.4.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 7$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{(x - 7) {(x - 7)}^{2}}{27}} - 1) + 10$

Step 5.3.4.1.8.4.3

Combine like factors.

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Step 5.3.4.1.8.4.3.1

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{(x - 7) {(x - 7)}^{2}}{27}} - 1) + 10$

Step 5.3.4.1.8.4.3.2

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{{(x - 7)}^{1 + 2}}{27}} - 1) + 10$

Step 5.3.4.1.8.4.3.3

Add $1$ and $2$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{{(x - 7)}^{3}}{27}} - 1) + 10$

Step 5.3.4.1.9

Rewrite $27$ as $3^{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{\frac{{(x - 7)}^{3}}{3^{3}}} - 1) + 10$

Step 5.3.4.1.10

Rewrite $\frac{{(x - 7)}^{3}}{3^{3}}$ as ${(\frac{x - 7}{3})}^{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\sqrt[3]{{(\frac{x - 7}{3})}^{3}} - 1) + 10$

Step 5.3.4.1.11

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 7}{3} - 1) + 10$

Step 5.3.4.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 7}{3} - 1 \cdot \frac{3}{3}) + 10$

Step 5.3.4.3

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

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 7}{3} + \frac{- 1 \cdot 3}{3}) + 10$

Step 5.3.4.4

Combine the numerators over the common denominator.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 7 - 1 \cdot 3}{3}) + 10$

Step 5.3.4.5

Simplify the numerator.

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Step 5.3.4.5.1

Multiply $- 1$ by $3$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 7 - 3}{3}) + 10$

Step 5.3.4.5.2

Subtract $3$ from $- 7$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 10}{3}) + 10$

Step 5.3.4.6

Cancel the common factor of $3$ .

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Step 5.3.4.6.1

Cancel the common factor.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = 3 (\frac{x - 10}{3}) + 10$

Step 5.3.4.6.2

Rewrite the expression.

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = x - 10 + 10$

Step 5.3.5

Combine the opposite terms in $x - 10 + 10$ .

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Step 5.3.5.1

Add $- 10$ and $10$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = x + 0$

Step 5.3.5.2

Add $x$ and $0$ .

$f (\frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}$ is the inverse of $f (x) = 3 (\sqrt[3]{x} - 1) + 10$ .

$f^{- 1} (x) = \frac{x^{3}}{27} - \frac{7 x^{2}}{9} + \frac{49 x}{9} - \frac{343}{27}$