Precalculus Examples

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

Step 1

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

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Step 1.1

Write $f (x) = 6 x^{\frac{1}{3}} - 2$ as an equation.

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

Step 1.2

Interchange the variables.

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

Step 1.3

Solve for $y$ .

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Step 1.3.1

Rewrite the equation as $6 y^{\frac{1}{3}} - 2 = x$ .

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

Step 1.3.2

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

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Step 1.3.2.1

Add $2$ to both sides of the equation.

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

Step 1.3.2.2

Add $x$ to both sides of the equation.

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

Step 1.3.3

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Step 1.3.4

Simplify the exponent.

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Step 1.3.4.1

Simplify the left side.

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Step 1.3.4.1.1

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

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Step 1.3.4.1.1.1

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

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

Step 1.3.4.1.1.2

Raise $6$ to the power of $3$ .

$216 {(y^{\frac{1}{3}})}^{3} = {(2 + x)}^{3}$

Step 1.3.4.1.1.3

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

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Step 1.3.4.1.1.3.1

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

$216 y^{\frac{1}{3} \cdot 3} = {(2 + x)}^{3}$

Step 1.3.4.1.1.3.2

Cancel the common factor of $3$ .

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Step 1.3.4.1.1.3.2.1

Cancel the common factor.

$216 y^{\frac{1}{3} \cdot 3} = {(2 + x)}^{3}$

Step 1.3.4.1.1.3.2.2

Rewrite the expression.

$216 y^{1} = {(2 + x)}^{3}$

Step 1.3.4.1.1.4

Simplify.

$216 y = {(2 + x)}^{3}$

Step 1.3.4.2

Simplify the right side.

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Step 1.3.4.2.1

Simplify ${(2 + x)}^{3}$ .

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Step 1.3.4.2.1.1

Use the Binomial Theorem.

$216 y = 2^{3} + 3 \cdot 2^{2} x + 3 \cdot 2 x^{2} + x^{3}$

Step 1.3.4.2.1.2

Simplify each term.

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Step 1.3.4.2.1.2.1

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

$216 y = 8 + 3 \cdot 2^{2} x + 3 \cdot 2 x^{2} + x^{3}$

Step 1.3.4.2.1.2.2

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

$216 y = 8 + 3 \cdot 4 x + 3 \cdot 2 x^{2} + x^{3}$

Step 1.3.4.2.1.2.3

Multiply $3$ by $4$ .

$216 y = 8 + 12 x + 3 \cdot 2 x^{2} + x^{3}$

Step 1.3.4.2.1.2.4

Multiply $3$ by $2$ .

$216 y = 8 + 12 x + 6 x^{2} + x^{3}$

Step 1.3.5

Divide each term in $216 y = 8 + 12 x + 6 x^{2} + x^{3}$ by $216$ and simplify.

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Step 1.3.5.1

Divide each term in $216 y = 8 + 12 x + 6 x^{2} + x^{3}$ by $216$ .

$\frac{216 y}{216} = \frac{8}{216} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.2

Simplify the left side.

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Step 1.3.5.2.1

Cancel the common factor of $216$ .

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Step 1.3.5.2.1.1

Cancel the common factor.

$\frac{216 y}{216} = \frac{8}{216} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{8}{216} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3

Simplify the right side.

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Step 1.3.5.3.1

Simplify each term.

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Step 1.3.5.3.1.1

Cancel the common factor of $8$ and $216$ .

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Step 1.3.5.3.1.1.1

Factor $8$ out of $8$ .

$y = \frac{8 (1)}{216} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.1.2

Cancel the common factors.

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Step 1.3.5.3.1.1.2.1

Factor $8$ out of $216$ .

$y = \frac{8 \cdot 1}{8 \cdot 27} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.1.2.2

Cancel the common factor.

$y = \frac{8 \cdot 1}{8 \cdot 27} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.1.2.3

Rewrite the expression.

$y = \frac{1}{27} + \frac{12 x}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.2

Cancel the common factor of $12$ and $216$ .

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Step 1.3.5.3.1.2.1

Factor $12$ out of $12 x$ .

$y = \frac{1}{27} + \frac{12 (x)}{216} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.2.2

Cancel the common factors.

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Step 1.3.5.3.1.2.2.1

Factor $12$ out of $216$ .

$y = \frac{1}{27} + \frac{12 x}{12 \cdot 18} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.2.2.2

Cancel the common factor.

$y = \frac{1}{27} + \frac{12 x}{12 \cdot 18} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.2.2.3

Rewrite the expression.

$y = \frac{1}{27} + \frac{x}{18} + \frac{6 x^{2}}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.3

Cancel the common factor of $6$ and $216$ .

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Step 1.3.5.3.1.3.1

Factor $6$ out of $6 x^{2}$ .

$y = \frac{1}{27} + \frac{x}{18} + \frac{6 (x^{2})}{216} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.3.2

Cancel the common factors.

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Step 1.3.5.3.1.3.2.1

Factor $6$ out of $216$ .

$y = \frac{1}{27} + \frac{x}{18} + \frac{6 x^{2}}{6 \cdot 36} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.3.2.2

Cancel the common factor.

$y = \frac{1}{27} + \frac{x}{18} + \frac{6 x^{2}}{6 \cdot 36} + \frac{x^{3}}{216}$

Step 1.3.5.3.1.3.2.3

Rewrite the expression.

$y = \frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}$

Step 1.4

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

$f^{- 1} (x) = \frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}$

Step 1.5

Verify if $f^{- 1} (x) = \frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}$ is the inverse of $f (x) = 6 x^{\frac{1}{3}} - 2$ .

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Step 1.5.1

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

Step 1.5.2

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

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Step 1.5.2.1

Set up the composite result function.

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

Step 1.5.2.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{6 x^{\frac{1}{3}} - 2}{18} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3

Simplify terms.

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Step 1.5.2.3.1

Simplify each term.

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Step 1.5.2.3.1.1

Factor $2$ out of $6 x^{\frac{1}{3}}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{2 (3 x^{\frac{1}{3}}) - 2}{18} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{2 (3 x^{\frac{1}{3}}) + 2 (- 1)}{18} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.3

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{2 (3 x^{\frac{1}{3}} - 1)}{18} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.4

Cancel the common factors.

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Step 1.5.2.3.1.4.1

Factor $2$ out of $18$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{2 (3 x^{\frac{1}{3}} - 1)}{2 (9)} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.4.2

Cancel the common factor.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{2 (3 x^{\frac{1}{3}} - 1)}{2 \cdot 9} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.4.3

Rewrite the expression.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.5

Simplify the numerator.

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Step 1.5.2.3.1.5.1

Factor $2$ out of $6 x^{\frac{1}{3}} - 2$ .

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Step 1.5.2.3.1.5.1.1

Factor $2$ out of $6 x^{\frac{1}{3}}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(2 (3 x^{\frac{1}{3}}) - 2)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.5.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(2 (3 x^{\frac{1}{3}}) + 2 (- 1))}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.5.1.3

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(2 (3 x^{\frac{1}{3}} - 1))}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.5.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{2^{2} {(3 x^{\frac{1}{3}} - 1)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.5.3

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{4 {(3 x^{\frac{1}{3}} - 1)}^{2}}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.6

Factor $4$ out of $4 {(3 x^{\frac{1}{3}} - 1)}^{2}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{4 ({(3 x^{\frac{1}{3}} - 1)}^{2})}{36} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.7

Cancel the common factors.

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Step 1.5.2.3.1.7.1

Factor $4$ out of $36$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{4 {(3 x^{\frac{1}{3}} - 1)}^{2}}{4 \cdot 9} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.7.2

Cancel the common factor.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{4 {(3 x^{\frac{1}{3}} - 1)}^{2}}{4 \cdot 9} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.7.3

Rewrite the expression.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{{(6 x^{\frac{1}{3}} - 2)}^{3}}{216}$

Step 1.5.2.3.1.8

Simplify the numerator.

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Step 1.5.2.3.1.8.1

Factor $2$ out of $6 x^{\frac{1}{3}} - 2$ .

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Step 1.5.2.3.1.8.1.1

Factor $2$ out of $6 x^{\frac{1}{3}}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{{(2 (3 x^{\frac{1}{3}}) - 2)}^{3}}{216}$

Step 1.5.2.3.1.8.1.2

Factor $2$ out of $- 2$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{{(2 (3 x^{\frac{1}{3}}) + 2 (- 1))}^{3}}{216}$

Step 1.5.2.3.1.8.1.3

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{{(2 (3 x^{\frac{1}{3}} - 1))}^{3}}{216}$

Step 1.5.2.3.1.8.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{2^{3} {(3 x^{\frac{1}{3}} - 1)}^{3}}{216}$

Step 1.5.2.3.1.8.3

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{8 {(3 x^{\frac{1}{3}} - 1)}^{3}}{216}$

Step 1.5.2.3.1.9

Factor $8$ out of $8 {(3 x^{\frac{1}{3}} - 1)}^{3}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{8 ({(3 x^{\frac{1}{3}} - 1)}^{3})}{216}$

Step 1.5.2.3.1.10

Cancel the common factors.

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Step 1.5.2.3.1.10.1

Factor $8$ out of $216$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{8 {(3 x^{\frac{1}{3}} - 1)}^{3}}{8 \cdot 27}$

Step 1.5.2.3.1.10.2

Cancel the common factor.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{8 {(3 x^{\frac{1}{3}} - 1)}^{3}}{8 \cdot 27}$

Step 1.5.2.3.1.10.3

Rewrite the expression.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{3 x^{\frac{1}{3}} - 1}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{2}}{9} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27}$

Step 1.5.2.3.2

Combine the numerators over the common denominator.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + {(3 x^{\frac{1}{3}} - 1)}^{2}}{9}$

Step 1.5.2.3.3

Simplify each term.

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Step 1.5.2.3.3.1

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + (3 x^{\frac{1}{3}} - 1) (3 x^{\frac{1}{3}} - 1)}{9}$

Step 1.5.2.3.3.2

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

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Step 1.5.2.3.3.2.1

Apply the distributive property.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1) - 1 (3 x^{\frac{1}{3}} - 1)}{9}$

Step 1.5.2.3.3.2.2

Apply the distributive property.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}} - 1)}{9}$

Step 1.5.2.3.3.2.3

Apply the distributive property.

Step 1.5.2.3.3.3

Simplify and combine like terms.

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Step 1.5.2.3.3.3.1

Simplify each term.

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Step 1.5.2.3.3.3.1.1

Rewrite using the commutative property of multiplication.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 \cdot (3 x^{\frac{1}{3}} x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.2

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

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Step 1.5.2.3.3.3.1.2.1

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 \cdot (3 (x^{\frac{1}{3}} x^{\frac{1}{3}})) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.2.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 \cdot (3 x^{\frac{1}{3} + \frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.2.3

Combine the numerators over the common denominator.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 \cdot (3 x^{\frac{1 + 1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.2.4

Add $1$ and $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 3 \cdot (3 x^{\frac{2}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.3

Multiply $3$ by $3$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 9 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.4

Multiply $- 1$ by $3$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.5

Multiply $3$ by $- 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 3 x^{\frac{1}{3}} - 1 \cdot - 1}{9}$

Step 1.5.2.3.3.3.1.6

Multiply $- 1$ by $- 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 3 x^{\frac{1}{3}} + 1}{9}$

Step 1.5.2.3.3.3.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} - 1 + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} + 1}{9}$

Step 1.5.2.3.4

Simplify by adding terms.

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Step 1.5.2.3.4.1

Combine the opposite terms in $3 x^{\frac{1}{3}} - 1 + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} + 1$ .

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Step 1.5.2.3.4.1.1

Add $- 1$ and $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} + 0}{9}$

Step 1.5.2.3.4.1.2

Add $3 x^{\frac{1}{3}} + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}}$ and $0$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}}}{9}$

Step 1.5.2.3.4.2

Subtract $6 x^{\frac{1}{3}}$ from $3 x^{\frac{1}{3}}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}}}{9}$

Step 1.5.2.3.5

Simplify each term.

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Step 1.5.2.3.5.1

Factor $3 x^{\frac{1}{3}}$ out of $9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}}$ .

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Step 1.5.2.3.5.1.1

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) - 3 x^{\frac{1}{3}}}{9}$

Step 1.5.2.3.5.1.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} (- 1)}{9}$

Step 1.5.2.3.5.1.3

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{9}$

Step 1.5.2.3.5.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 (x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1))}{9}$

Step 1.5.2.3.5.3

Cancel the common factors.

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Step 1.5.2.3.5.3.1

Factor $3$ out of $9$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 (x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1))}{3 \cdot 3}$

Step 1.5.2.3.5.3.2

Cancel the common factor.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{3 (x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1))}{3 \cdot 3}$

Step 1.5.2.3.5.3.3

Rewrite the expression.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1}{27} + \frac{{(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.6

Combine the numerators over the common denominator.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1 + {(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7

Simplify each term.

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Step 1.5.2.3.7.1

Simplify the numerator.

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Step 1.5.2.3.7.1.1

Rewrite $1$ as $1^{3}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{1^{3} + {(3 x^{\frac{1}{3}} - 1)}^{3}}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = 1$ and $b = 3 x^{\frac{1}{3}} - 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{(1 + 3 x^{\frac{1}{3}} - 1) (1^{2} - 1 (3 x^{\frac{1}{3}} - 1) + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3

Simplify.

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Step 1.5.2.3.7.1.3.1

Subtract $1$ from $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{(3 x^{\frac{1}{3}} + 0) (1^{2} - 1 (3 x^{\frac{1}{3}} - 1) + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1^{2} - 1 (3 x^{\frac{1}{3}} - 1) + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.3

One to any power is one.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 1 (3 x^{\frac{1}{3}} - 1) + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.4

Apply the distributive property.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1 + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.5

Multiply $3$ by $- 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} - 1 \cdot - 1 + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.6

Multiply $- 1$ by $- 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + {(3 x^{\frac{1}{3}} - 1)}^{2})}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.7

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + (3 x^{\frac{1}{3}} - 1) (3 x^{\frac{1}{3}} - 1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.8

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

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Step 1.5.2.3.7.1.3.8.1

Apply the distributive property.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1) - 1 (3 x^{\frac{1}{3}} - 1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.8.2

Apply the distributive property.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}} - 1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.8.3

Apply the distributive property.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 x^{\frac{1}{3}} (3 x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9

Simplify and combine like terms.

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Step 1.5.2.3.7.1.3.9.1

Simplify each term.

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Step 1.5.2.3.7.1.3.9.1.1

Rewrite using the commutative property of multiplication.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 \cdot (3 x^{\frac{1}{3}} x^{\frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.2

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

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Step 1.5.2.3.7.1.3.9.1.2.1

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 \cdot (3 (x^{\frac{1}{3}} x^{\frac{1}{3}})) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.2.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 \cdot (3 x^{\frac{1}{3} + \frac{1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.2.3

Combine the numerators over the common denominator.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 \cdot (3 x^{\frac{1 + 1}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.2.4

Add $1$ and $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 3 \cdot (3 x^{\frac{2}{3}}) + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.3

Multiply $3$ by $3$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 9 x^{\frac{2}{3}} + 3 x^{\frac{1}{3}} \cdot - 1 - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.4

Multiply $- 1$ by $3$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 1 (3 x^{\frac{1}{3}}) - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.5

Multiply $3$ by $- 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 3 x^{\frac{1}{3}} - 1 \cdot - 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.1.6

Multiply $- 1$ by $- 1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 9 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} - 3 x^{\frac{1}{3}} + 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.9.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (1 - 3 x^{\frac{1}{3}} + 1 + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} + 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.10

Add $1$ and $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (- 3 x^{\frac{1}{3}} + 2 + 9 x^{\frac{2}{3}} - 6 x^{\frac{1}{3}} + 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.11

Subtract $6 x^{\frac{1}{3}}$ from $- 3 x^{\frac{1}{3}}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (- 9 x^{\frac{1}{3}} + 2 + 9 x^{\frac{2}{3}} + 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.12

Add $2$ and $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (- 9 x^{\frac{1}{3}} + 9 x^{\frac{2}{3}} + 3)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.13

Reorder terms.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (9 x^{\frac{2}{3}} - 9 x^{\frac{1}{3}} + 3)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.14

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

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Step 1.5.2.3.7.1.3.14.1

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (3 (3 x^{\frac{2}{3}}) - 9 x^{\frac{1}{3}} + 3)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.14.2

Factor $3$ out of $- 9 x^{\frac{1}{3}}$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (3 (3 x^{\frac{2}{3}}) + 3 (- 3 x^{\frac{1}{3}}) + 3)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.14.3

Factor $3$ out of $3$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (3 (3 x^{\frac{2}{3}}) + 3 (- 3 x^{\frac{1}{3}}) + 3 (1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.14.4

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (3 (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}}) + 3 (1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.14.5

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} (3 (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{3 x^{\frac{1}{3}} \cdot (3 (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.1.3.15

Multiply $3$ by $3$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{9 x^{\frac{1}{3}} (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1)}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.2

Factor $9$ out of $9 x^{\frac{1}{3}} (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1)$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{9 (x^{\frac{1}{3}} (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1))}{27} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.3

Cancel the common factors.

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Step 1.5.2.3.7.3.1

Factor $9$ out of $27$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{9 (x^{\frac{1}{3}} (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1))}{9 \cdot 3} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.3.2

Cancel the common factor.

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{9 (x^{\frac{1}{3}} (3 x^{\frac{2}{3}} - 3 x^{\frac{1}{3}} + 1))}{9 \cdot 3} + \frac{x^{\frac{1}{3}} (3 x^{\frac{1}{3}} - 1)}{3}$

Step 1.5.2.3.7.3.3

Rewrite the expression.

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

Step 1.5.2.3.8

Combine the numerators over the common denominator.

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

Step 1.5.2.3.9

Simplify each term.

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Step 1.5.2.3.9.1

Apply the distributive property.

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

Step 1.5.2.3.9.2

Simplify.

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Step 1.5.2.3.9.2.1

Rewrite using the commutative property of multiplication.

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

Step 1.5.2.3.9.2.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.2.3.9.2.3

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

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

Step 1.5.2.3.9.3

Simplify each term.

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Step 1.5.2.3.9.3.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.5.2.3.9.3.1.1

Move $x^{\frac{2}{3}}$ .

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

Step 1.5.2.3.9.3.1.2

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

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

Step 1.5.2.3.9.3.1.3

Combine the numerators over the common denominator.

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

Step 1.5.2.3.9.3.1.4

Add $2$ and $1$ .

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

Step 1.5.2.3.9.3.1.5

Divide $3$ by $3$ .

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

Step 1.5.2.3.9.3.2

Simplify $3 x^{1}$ .

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

Step 1.5.2.3.9.3.3

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

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Step 1.5.2.3.9.3.3.1

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

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

Step 1.5.2.3.9.3.3.2

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

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

Step 1.5.2.3.9.3.3.3

Combine the numerators over the common denominator.

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

Step 1.5.2.3.9.3.3.4

Add $1$ and $1$ .

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

Step 1.5.2.4

Simplify the numerator.

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Step 1.5.2.4.1

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

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Step 1.5.2.4.1.1

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

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

Step 1.5.2.4.1.2

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

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

Step 1.5.2.4.1.3

Multiply by $1$ .

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

Step 1.5.2.4.1.4

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

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

Step 1.5.2.4.1.5

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

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

Step 1.5.2.4.1.6

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

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

Step 1.5.2.4.2

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

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{x^{\frac{1}{3}} (3 x^{\frac{2}{3}} + 0 + 1 - 1)}{3}$

Step 1.5.2.4.3

Add $3 x^{\frac{2}{3}}$ and $0$ .

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

Step 1.5.2.4.4

Subtract $1$ from $1$ .

$f^{- 1} (6 x^{\frac{1}{3}} - 2) = \frac{x^{\frac{1}{3}} (3 x^{\frac{2}{3}} + 0)}{3}$

Step 1.5.2.4.5

Add $3 x^{\frac{2}{3}}$ and $0$ .

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

Step 1.5.2.4.6

Combine exponents.

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Step 1.5.2.4.6.1

Multiply $x^{\frac{1}{3}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.5.2.4.6.1.1

Move $x^{\frac{2}{3}}$ .

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

Step 1.5.2.4.6.1.2

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

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

Step 1.5.2.4.6.1.3

Combine the numerators over the common denominator.

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

Step 1.5.2.4.6.1.4

Add $2$ and $1$ .

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

Step 1.5.2.4.6.1.5

Divide $3$ by $3$ .

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

Step 1.5.2.4.6.2

Simplify $x^{1} \cdot 3$ .

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

Step 1.5.2.5

Cancel the common factor of $3$ .

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Step 1.5.2.5.1

Cancel the common factor.

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

Step 1.5.2.5.2

Divide $x$ by $1$ .

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

Step 1.5.3

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

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Step 1.5.3.1

Set up the composite result function.

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

Step 1.5.3.2

Evaluate $f (\frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}) = 6 {(\frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216})}^{\frac{1}{3}} - 2$

Step 1.5.3.3

Move $\frac{1}{27}$ .

$f (\frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}) = 6 {(\frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216} + \frac{1}{27})}^{\frac{1}{3}} - 2$

Step 1.5.3.4

Move $\frac{x}{18}$ .

$f (\frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}) = 6 {(\frac{x^{2}}{36} + \frac{x^{3}}{216} + \frac{x}{18} + \frac{1}{27})}^{\frac{1}{3}} - 2$

Step 1.5.3.5

Reorder $\frac{x^{2}}{36}$ and $\frac{x^{3}}{216}$ .

$f (\frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}) = 6 {(\frac{x^{3}}{216} + \frac{x^{2}}{36} + \frac{x}{18} + \frac{1}{27})}^{\frac{1}{3}} - 2$

Step 1.5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}$ is the inverse of $f (x) = 6 x^{\frac{1}{3}} - 2$ .

$f^{- 1} (x) = \frac{1}{27} + \frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216}$

Step 2

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Step 2.1

Move $\frac{1}{27}$ .

$\frac{x}{18} + \frac{x^{2}}{36} + \frac{x^{3}}{216} + \frac{1}{27}$

Step 2.2

Move $\frac{x}{18}$ .

$\frac{x^{2}}{36} + \frac{x^{3}}{216} + \frac{x}{18} + \frac{1}{27}$

Step 2.3

Reorder $\frac{x^{2}}{36}$ and $\frac{x^{3}}{216}$ .

$\frac{x^{3}}{216} + \frac{x^{2}}{36} + \frac{x}{18} + \frac{1}{27}$