Algebra Examples

$f (x) = 5 (\frac{\sqrt[3]{x}}{6} + 10)$

Step 1

Write $f (x) = 5 (\frac{\sqrt[3]{x}}{6} + 10)$ as an equation.

$y = 5 (\frac{\sqrt[3]{x}}{6} + 10)$

Step 2

Interchange the variables.

$x = 5 (\frac{\sqrt[3]{y}}{6} + 10)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $5 (\frac{\sqrt[3]{y}}{6} + 10) = x$ .

$5 (\frac{\sqrt[3]{y}}{6} + 10) = x$

Step 3.2

Divide each term in $5 (\frac{\sqrt[3]{y}}{6} + 10) = x$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 (\frac{\sqrt[3]{y}}{6} + 10) = x$ by $5$ .

$\frac{5 (\frac{\sqrt[3]{y}}{6} + 10)}{5} = \frac{x}{5}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 (\frac{\sqrt[3]{y}}{6} + 10)}{5} = \frac{x}{5}$

Step 3.2.2.1.2

Divide $\frac{\sqrt[3]{y}}{6} + 10$ by $1$ .

$\frac{\sqrt[3]{y}}{6} + 10 = \frac{x}{5}$

Step 3.3

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

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Step 3.3.1

Subtract $10$ from both sides of the equation.

$\frac{\sqrt[3]{y}}{6} = \frac{x}{5} - 10$

Step 3.3.2

Multiply both sides of the equation by $6$ .

$6 \frac{\sqrt[3]{y}}{6} = 6 (\frac{x}{5} - 10)$

Step 3.3.3

Simplify both sides of the equation.

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Step 3.3.3.1

Simplify the left side.

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Step 3.3.3.1.1

Cancel the common factor of $6$ .

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Step 3.3.3.1.1.1

Cancel the common factor.

$6 \frac{\sqrt[3]{y}}{6} = 6 (\frac{x}{5} - 10)$

Step 3.3.3.1.1.2

Rewrite the expression.

$\sqrt[3]{y} = 6 (\frac{x}{5} - 10)$

Step 3.3.3.2

Simplify the right side.

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Step 3.3.3.2.1

Simplify $6 (\frac{x}{5} - 10)$ .

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Step 3.3.3.2.1.1

Apply the distributive property.

$\sqrt[3]{y} = 6 \frac{x}{5} + 6 \cdot - 10$

Step 3.3.3.2.1.2

Combine $6$ and $\frac{x}{5}$ .

$\sqrt[3]{y} = \frac{6 x}{5} + 6 \cdot - 10$

Step 3.3.3.2.1.3

Multiply $6$ by $- 10$ .

$\sqrt[3]{y} = \frac{6 x}{5} - 60$

Step 3.4

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

${\sqrt[3]{y}}^{3} = {(\frac{6 x}{5} - 60)}^{3}$

Step 3.5

Simplify each side of the equation.

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Step 3.5.1

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

${(y^{\frac{1}{3}})}^{3} = {(\frac{6 x}{5} - 60)}^{3}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

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

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Step 3.5.2.1.1

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

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Step 3.5.2.1.1.1

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

$y^{\frac{1}{3} \cdot 3} = {(\frac{6 x}{5} - 60)}^{3}$

Step 3.5.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.5.2.1.1.2.1

Cancel the common factor.

$y^{\frac{1}{3} \cdot 3} = {(\frac{6 x}{5} - 60)}^{3}$

Step 3.5.2.1.1.2.2

Rewrite the expression.

$y^{1} = {(\frac{6 x}{5} - 60)}^{3}$

Step 3.5.2.1.2

Simplify.

$y = {(\frac{6 x}{5} - 60)}^{3}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify ${(\frac{6 x}{5} - 60)}^{3}$ .

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Step 3.5.3.1.1

Use the Binomial Theorem.

$y = {(\frac{6 x}{5})}^{3} + 3 {(\frac{6 x}{5})}^{2} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2

Simplify each term.

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Step 3.5.3.1.2.1

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

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Step 3.5.3.1.2.1.1

Apply the product rule to $\frac{6 x}{5}$ .

$y = \frac{{(6 x)}^{3}}{5^{3}} + 3 {(\frac{6 x}{5})}^{2} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.1.2

Apply the product rule to $6 x$ .

$y = \frac{6^{3} x^{3}}{5^{3}} + 3 {(\frac{6 x}{5})}^{2} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.2

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

$y = \frac{216 x^{3}}{5^{3}} + 3 {(\frac{6 x}{5})}^{2} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.3

Raise $5$ to the power of $3$ .

$y = \frac{216 x^{3}}{125} + 3 {(\frac{6 x}{5})}^{2} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.4

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

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Step 3.5.3.1.2.4.1

Apply the product rule to $\frac{6 x}{5}$ .

$y = \frac{216 x^{3}}{125} + 3 \frac{{(6 x)}^{2}}{5^{2}} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.4.2

Apply the product rule to $6 x$ .

$y = \frac{216 x^{3}}{125} + 3 \frac{6^{2} x^{2}}{5^{2}} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.5

Raise $6$ to the power of $2$ .

$y = \frac{216 x^{3}}{125} + 3 \frac{36 x^{2}}{5^{2}} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.6

Raise $5$ to the power of $2$ .

$y = \frac{216 x^{3}}{125} + 3 \frac{36 x^{2}}{25} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.7

Multiply $3 \frac{36 x^{2}}{25}$ .

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Step 3.5.3.1.2.7.1

Combine $3$ and $\frac{36 x^{2}}{25}$ .

$y = \frac{216 x^{3}}{125} + \frac{3 (36 x^{2})}{25} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.7.2

Multiply $36$ by $3$ .

$y = \frac{216 x^{3}}{125} + \frac{108 x^{2}}{25} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.8

Cancel the common factor of $5$ .

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Step 3.5.3.1.2.8.1

Factor $5$ out of $25$ .

$y = \frac{216 x^{3}}{125} + \frac{108 x^{2}}{5 (5)} \cdot - 60 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.8.2

Factor $5$ out of $- 60$ .

$y = \frac{216 x^{3}}{125} + \frac{108 x^{2}}{5 \cdot 5} \cdot (5 \cdot - 12) + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.8.3

Cancel the common factor.

$y = \frac{216 x^{3}}{125} + \frac{108 x^{2}}{5 \cdot 5} \cdot (5 \cdot - 12) + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.8.4

Rewrite the expression.

$y = \frac{216 x^{3}}{125} + \frac{108 x^{2}}{5} \cdot - 12 + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.9

Combine $\frac{108 x^{2}}{5}$ and $- 12$ .

$y = \frac{216 x^{3}}{125} + \frac{108 x^{2} \cdot - 12}{5} + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.10

Multiply $- 12$ by $108$ .

$y = \frac{216 x^{3}}{125} + \frac{- 1296 x^{2}}{5} + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.11

Move the negative in front of the fraction.

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 3 \frac{6 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.12

Multiply $3 \frac{6 x}{5}$ .

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Step 3.5.3.1.2.12.1

Combine $3$ and $\frac{6 x}{5}$ .

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + \frac{3 (6 x)}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.12.2

Multiply $6$ by $3$ .

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + \frac{18 x}{5} {(- 60)}^{2} + {(- 60)}^{3}$

Step 3.5.3.1.2.13

Raise $- 60$ to the power of $2$ .

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + \frac{18 x}{5} \cdot 3600 + {(- 60)}^{3}$

Step 3.5.3.1.2.14

Cancel the common factor of $5$ .

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Step 3.5.3.1.2.14.1

Factor $5$ out of $3600$ .

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + \frac{18 x}{5} \cdot (5 (720)) + {(- 60)}^{3}$

Step 3.5.3.1.2.14.2

Cancel the common factor.

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + \frac{18 x}{5} \cdot (5 \cdot 720) + {(- 60)}^{3}$

Step 3.5.3.1.2.14.3

Rewrite the expression.

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 18 x \cdot 720 + {(- 60)}^{3}$

Step 3.5.3.1.2.15

Multiply $720$ by $18$ .

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x + {(- 60)}^{3}$

Step 3.5.3.1.2.16

Raise $- 60$ to the power of $3$ .

$y = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000$

Step 4

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

$f^{- 1} (x) = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000$

Step 5

Verify if $f^{- 1} (x) = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000$ is the inverse of $f (x) = 5 (\frac{\sqrt[3]{x}}{6} + 10)$ .

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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 $f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{3}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Apply the product rule to $5 (\frac{\sqrt[3]{x}}{6} + 10)$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 \cdot (5^{3} {(\frac{\sqrt[3]{x}}{6} + 10)}^{3})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.2

Raise $5$ to the power of $3$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 \cdot (125 {(\frac{\sqrt[3]{x}}{6} + 10)}^{3})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.3

To write $10$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 \cdot (125 {(\frac{\sqrt[3]{x}}{6} + 10 \cdot \frac{6}{6})}^{3})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.4

Combine $10$ and $\frac{6}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 \cdot (125 {(\frac{\sqrt[3]{x}}{6} + \frac{10 \cdot 6}{6})}^{3})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.5

Combine the numerators over the common denominator.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 \cdot (125 {(\frac{\sqrt[3]{x} + 10 \cdot 6}{6})}^{3})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.6

Multiply $10$ by $6$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{216 \cdot (125 {(\frac{\sqrt[3]{x} + 60}{6})}^{3})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.7

Multiply $216$ by $125$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{27000 {(\frac{\sqrt[3]{x} + 60}{6})}^{3}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.8

Apply the product rule to $\frac{\sqrt[3]{x} + 60}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{27000 (\frac{{(\sqrt[3]{x} + 60)}^{3}}{6^{3}})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.1.9

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{27000 (\frac{{(\sqrt[3]{x} + 60)}^{3}}{216})}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.2

Combine $27000$ and $\frac{{(\sqrt[3]{x} + 60)}^{3}}{216}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{\frac{27000 {(\sqrt[3]{x} + 60)}^{3}}{216}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.3

Reduce the expression by cancelling the common factors.

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Step 5.2.3.3.1

Reduce the expression $\frac{27000 {(\sqrt[3]{x} + 60)}^{3}}{216}$ by cancelling the common factors.

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Step 5.2.3.3.1.1

Factor $216$ out of $27000 {(\sqrt[3]{x} + 60)}^{3}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{\frac{216 (125 {(\sqrt[3]{x} + 60)}^{3})}{216}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.3.1.2

Factor $216$ out of $216$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{\frac{216 (125 {(\sqrt[3]{x} + 60)}^{3})}{216 (1)}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.3.1.3

Cancel the common factor.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{\frac{216 (125 {(\sqrt[3]{x} + 60)}^{3})}{216 \cdot 1}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.3.1.4

Rewrite the expression.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{\frac{125 {(\sqrt[3]{x} + 60)}^{3}}{1}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.3.2

Divide $125 {(\sqrt[3]{x} + 60)}^{3}$ by $1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{125 {(\sqrt[3]{x} + 60)}^{3}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.4

Cancel the common factor of $125$ .

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Step 5.2.3.4.1

Cancel the common factor.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = \frac{125 {(\sqrt[3]{x} + 60)}^{3}}{125} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.4.2

Divide ${(\sqrt[3]{x} + 60)}^{3}$ by $1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = {(\sqrt[3]{x} + 60)}^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.5

Use the Binomial Theorem.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = {\sqrt[3]{x}}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 60 + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6

Simplify each term.

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Step 5.2.3.6.1

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

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Step 5.2.3.6.1.1

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = {(x^{\frac{1}{3}})}^{3} + 3 {\sqrt[3]{x}}^{2} \cdot 60 + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.1.2

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x^{\frac{1}{3} \cdot 3} + 3 {\sqrt[3]{x}}^{2} \cdot 60 + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.1.3

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x^{\frac{3}{3}} + 3 {\sqrt[3]{x}}^{2} \cdot 60 + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.1.4

Cancel the common factor of $3$ .

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Step 5.2.3.6.1.4.1

Cancel the common factor.

Step 5.2.3.6.1.4.2

Rewrite the expression.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 3 {\sqrt[3]{x}}^{2} \cdot 60 + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.1.5

Simplify.

Step 5.2.3.6.2

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 3 \sqrt[3]{x^{2}} \cdot 60 + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.3

Multiply $60$ by $3$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 60^{2} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.4

Raise $60$ to the power of $2$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 3 \sqrt[3]{x} \cdot 3600 + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.5

Multiply $3600$ by $3$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 60^{3} - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.6.6

Raise $60$ to the power of $3$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 {(5 (\frac{\sqrt[3]{x}}{6} + 10))}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7

Simplify the numerator.

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Step 5.2.3.7.1

Apply the product rule to $5 (\frac{\sqrt[3]{x}}{6} + 10)$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 \cdot (5^{2} {(\frac{\sqrt[3]{x}}{6} + 10)}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.2

Raise $5$ to the power of $2$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 \cdot (25 {(\frac{\sqrt[3]{x}}{6} + 10)}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.3

To write $10$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 \cdot (25 {(\frac{\sqrt[3]{x}}{6} + 10 \cdot \frac{6}{6})}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.4

Combine $10$ and $\frac{6}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 \cdot (25 {(\frac{\sqrt[3]{x}}{6} + \frac{10 \cdot 6}{6})}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.5

Combine the numerators over the common denominator.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 \cdot (25 {(\frac{\sqrt[3]{x} + 10 \cdot 6}{6})}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.6

Multiply $10$ by $6$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{1296 \cdot (25 {(\frac{\sqrt[3]{x} + 60}{6})}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.7

Multiply $1296$ by $25$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{32400 {(\frac{\sqrt[3]{x} + 60}{6})}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.8

Apply the product rule to $\frac{\sqrt[3]{x} + 60}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{32400 (\frac{{(\sqrt[3]{x} + 60)}^{2}}{6^{2}})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.7.9

Raise $6$ to the power of $2$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{32400 (\frac{{(\sqrt[3]{x} + 60)}^{2}}{36})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.8

Combine $32400$ and $\frac{{(\sqrt[3]{x} + 60)}^{2}}{36}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{\frac{32400 {(\sqrt[3]{x} + 60)}^{2}}{36}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.9

Reduce the expression by cancelling the common factors.

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Step 5.2.3.9.1

Reduce the expression $\frac{32400 {(\sqrt[3]{x} + 60)}^{2}}{36}$ by cancelling the common factors.

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Step 5.2.3.9.1.1

Factor $36$ out of $32400 {(\sqrt[3]{x} + 60)}^{2}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{\frac{36 (900 {(\sqrt[3]{x} + 60)}^{2})}{36}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.9.1.2

Factor $36$ out of $36$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{\frac{36 (900 {(\sqrt[3]{x} + 60)}^{2})}{36 (1)}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.9.1.3

Cancel the common factor.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{\frac{36 (900 {(\sqrt[3]{x} + 60)}^{2})}{36 \cdot 1}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.9.1.4

Rewrite the expression.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{\frac{900 {(\sqrt[3]{x} + 60)}^{2}}{1}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.9.2

Divide $900 {(\sqrt[3]{x} + 60)}^{2}$ by $1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{900 {(\sqrt[3]{x} + 60)}^{2}}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.10

Cancel the common factor of $900$ and $5$ .

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Step 5.2.3.10.1

Factor $5$ out of $900 {(\sqrt[3]{x} + 60)}^{2}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{5 (180 {(\sqrt[3]{x} + 60)}^{2})}{5} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.10.2

Cancel the common factors.

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Step 5.2.3.10.2.1

Factor $5$ out of $5$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{5 (180 {(\sqrt[3]{x} + 60)}^{2})}{5 (1)} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.10.2.2

Cancel the common factor.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{5 (180 {(\sqrt[3]{x} + 60)}^{2})}{5 \cdot 1} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.10.2.3

Rewrite the expression.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - \frac{180 {(\sqrt[3]{x} + 60)}^{2}}{1} + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.10.2.4

Divide $180 {(\sqrt[3]{x} + 60)}^{2}$ by $1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 {(\sqrt[3]{x} + 60)}^{2}) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.11

Rewrite ${(\sqrt[3]{x} + 60)}^{2}$ as $(\sqrt[3]{x} + 60) (\sqrt[3]{x} + 60)$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 ((\sqrt[3]{x} + 60) (\sqrt[3]{x} + 60))) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.12

Expand $(\sqrt[3]{x} + 60) (\sqrt[3]{x} + 60)$ using the FOIL Method.

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Step 5.2.3.12.1

Apply the distributive property.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x} (\sqrt[3]{x} + 60) + 60 (\sqrt[3]{x} + 60))) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.12.2

Apply the distributive property.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 60 + 60 (\sqrt[3]{x} + 60))) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.12.3

Apply the distributive property.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x} \sqrt[3]{x} + \sqrt[3]{x} \cdot 60 + 60 \sqrt[3]{x} + 60 \cdot 60)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.13

Simplify and combine like terms.

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Step 5.2.3.13.1

Simplify each term.

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Step 5.2.3.13.1.1

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

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Step 5.2.3.13.1.1.1

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

Step 5.2.3.13.1.1.2

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

Step 5.2.3.13.1.1.3

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 ({\sqrt[3]{x}}^{1 + 1} + \sqrt[3]{x} \cdot 60 + 60 \sqrt[3]{x} + 60 \cdot 60)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.13.1.1.4

Add $1$ and $1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 ({\sqrt[3]{x}}^{2} + \sqrt[3]{x} \cdot 60 + 60 \sqrt[3]{x} + 60 \cdot 60)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.13.1.2

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

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x^{2}} + \sqrt[3]{x} \cdot 60 + 60 \sqrt[3]{x} + 60 \cdot 60)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.13.1.3

Move $60$ to the left of $\sqrt[3]{x}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x^{2}} + 60 \cdot \sqrt[3]{x} + 60 \sqrt[3]{x} + 60 \cdot 60)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.13.1.4

Multiply $60$ by $60$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x^{2}} + 60 \sqrt[3]{x} + 60 \sqrt[3]{x} + 3600)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.13.2

Add $60 \sqrt[3]{x}$ and $60 \sqrt[3]{x}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 (\sqrt[3]{x^{2}} + 120 \sqrt[3]{x} + 3600)) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.14

Apply the distributive property.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 \sqrt[3]{x^{2}} + 180 (120 \sqrt[3]{x}) + 180 \cdot 3600) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.15

Simplify.

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Step 5.2.3.15.1

Multiply $120$ by $180$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 \sqrt[3]{x^{2}} + 21600 \sqrt[3]{x} + 180 \cdot 3600) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.15.2

Multiply $180$ by $3600$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 \sqrt[3]{x^{2}} + 21600 \sqrt[3]{x} + 648000) + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.16

Apply the distributive property.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - (180 \sqrt[3]{x^{2}}) - (21600 \sqrt[3]{x}) - 1 \cdot 648000 + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.17

Simplify.

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Step 5.2.3.17.1

Multiply $180$ by $- 1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - (21600 \sqrt[3]{x}) - 1 \cdot 648000 + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.17.2

Multiply $21600$ by $- 1$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 1 \cdot 648000 + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.17.3

Multiply $- 1$ by $648000$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 12960 (5 (\frac{\sqrt[3]{x}}{6} + 10)) - 216000$

Step 5.2.3.18

Apply the distributive property.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 12960 (5 (\frac{\sqrt[3]{x}}{6}) + 5 \cdot 10) - 216000$

Step 5.2.3.19

Combine $5$ and $\frac{\sqrt[3]{x}}{6}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 12960 (\frac{5 \sqrt[3]{x}}{6} + 5 \cdot 10) - 216000$

Step 5.2.3.20

Multiply $5$ by $10$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 12960 (\frac{5 \sqrt[3]{x}}{6} + 50) - 216000$

Step 5.2.3.21

Apply the distributive property.

Step 5.2.3.22

Cancel the common factor of $6$ .

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Step 5.2.3.22.1

Factor $6$ out of $12960$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 6 (2160) (\frac{5 \sqrt[3]{x}}{6}) + 12960 \cdot 50 - 216000$

Step 5.2.3.22.2

Cancel the common factor.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 6 \cdot (2160 (\frac{5 \sqrt[3]{x}}{6})) + 12960 \cdot 50 - 216000$

Step 5.2.3.22.3

Rewrite the expression.

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 2160 (5 \sqrt[3]{x}) + 12960 \cdot 50 - 216000$

Step 5.2.3.23

Multiply $5$ by $2160$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 10800 \sqrt[3]{x} + 12960 \cdot 50 - 216000$

Step 5.2.3.24

Multiply $12960$ by $50$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 10800 \sqrt[3]{x} + 648000 - 216000$

Step 5.2.4

Simplify by adding terms.

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Step 5.2.4.1

Combine the opposite terms in $x + 180 \sqrt[3]{x^{2}} + 10800 \sqrt[3]{x} + 216000 - 180 \sqrt[3]{x^{2}} - 21600 \sqrt[3]{x} - 648000 + 10800 \sqrt[3]{x} + 648000 - 216000$ .

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Step 5.2.4.1.1

Subtract $180 \sqrt[3]{x^{2}}$ from $180 \sqrt[3]{x^{2}}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 0 + 10800 \sqrt[3]{x} + 216000 - 21600 \sqrt[3]{x} - 648000 + 10800 \sqrt[3]{x} + 648000 - 216000$

Step 5.2.4.1.2

Add $x$ and $0$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 10800 \sqrt[3]{x} + 216000 - 21600 \sqrt[3]{x} - 648000 + 10800 \sqrt[3]{x} + 648000 - 216000$

Step 5.2.4.1.3

Add $- 648000$ and $648000$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 10800 \sqrt[3]{x} + 216000 - 21600 \sqrt[3]{x} + 0 + 10800 \sqrt[3]{x} - 216000$

Step 5.2.4.1.4

Add $x + 10800 \sqrt[3]{x} + 216000 - 21600 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 10800 \sqrt[3]{x} + 216000 - 21600 \sqrt[3]{x} + 10800 \sqrt[3]{x} - 216000$

Step 5.2.4.1.5

Subtract $216000$ from $216000$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 10800 \sqrt[3]{x} + 0 - 21600 \sqrt[3]{x} + 10800 \sqrt[3]{x}$

Step 5.2.4.1.6

Add $x + 10800 \sqrt[3]{x}$ and $0$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 10800 \sqrt[3]{x} - 21600 \sqrt[3]{x} + 10800 \sqrt[3]{x}$

Step 5.2.4.2

Subtract $21600 \sqrt[3]{x}$ from $10800 \sqrt[3]{x}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x - 10800 \sqrt[3]{x} + 10800 \sqrt[3]{x}$

Step 5.2.4.3

Combine the opposite terms in $x - 10800 \sqrt[3]{x} + 10800 \sqrt[3]{x}$ .

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Step 5.2.4.3.1

Add $- 10800 \sqrt[3]{x}$ and $10800 \sqrt[3]{x}$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x + 0$

Step 5.2.4.3.2

Add $x$ and $0$ .

$f^{- 1} (5 (\frac{\sqrt[3]{x}}{6} + 10)) = x$

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 $f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000}}{6} + 10)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Simplify the numerator.

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Step 5.3.3.1.1

To write $- \frac{1296 x^{2}}{5}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} \cdot \frac{25}{25} + 12960 x - 216000}}{6} + 10)$

Step 5.3.3.1.2

Write each expression with a common denominator of $125$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.1.2.1

Multiply $\frac{1296 x^{2}}{5}$ by $\frac{25}{25}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3}}{125} - \frac{1296 x^{2} \cdot 25}{5 \cdot 25} + 12960 x - 216000}}{6} + 10)$

Step 5.3.3.1.2.2

Multiply $5$ by $25$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3}}{125} - \frac{1296 x^{2} \cdot 25}{125} + 12960 x - 216000}}{6} + 10)$

Step 5.3.3.1.3

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25}{125} + 12960 x - 216000}}{6} + 10)$

Step 5.3.3.1.4

To write $12960 x$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25}{125} + 12960 x \cdot \frac{125}{125} - 216000}}{6} + 10)$

Step 5.3.3.1.5

Combine $12960 x$ and $\frac{125}{125}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25}{125} + \frac{12960 x \cdot 125}{125} - 216000}}{6} + 10)$

Step 5.3.3.1.6

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25 + 12960 x \cdot 125}{125} - 216000}}{6} + 10)$

Step 5.3.3.1.7

To write $- 216000$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25 + 12960 x \cdot 125}{125} - 216000 \cdot \frac{125}{125}}}{6} + 10)$

Step 5.3.3.1.8

Combine $- 216000$ and $\frac{125}{125}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25 + 12960 x \cdot 125}{125} + \frac{- 216000 \cdot 125}{125}}}{6} + 10)$

Step 5.3.3.1.9

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 x^{3} - 1296 x^{2} \cdot 25 + 12960 x \cdot 125 - 216000 \cdot 125}{125}}}{6} + 10)$

Step 5.3.3.1.10

Rewrite $\frac{216 x^{3} - 1296 x^{2} \cdot 25 + 12960 x \cdot 125 - 216000 \cdot 125}{125}$ in a factored form.

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Step 5.3.3.1.10.1

Factor $216$ out of $216 x^{3} - 1296 x^{2} \cdot 25 + 12960 x \cdot 125 - 216000 \cdot 125$ .

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Step 5.3.3.1.10.1.1

Factor $216$ out of $216 x^{3}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3}) - 1296 x^{2} \cdot 25 + 12960 x \cdot 125 - 216000 \cdot 125}{125}}}{6} + 10)$

Step 5.3.3.1.10.1.2

Factor $216$ out of $- 1296 x^{2} \cdot 25$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3}) + 216 (- 6 x^{2} \cdot 25) + 12960 x \cdot 125 - 216000 \cdot 125}{125}}}{6} + 10)$

Step 5.3.3.1.10.1.3

Factor $216$ out of $12960 x \cdot 125$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3}) + 216 (- 6 x^{2} \cdot 25) + 216 (60 x \cdot 125) - 216000 \cdot 125}{125}}}{6} + 10)$

Step 5.3.3.1.10.1.4

Factor $216$ out of $- 216000 \cdot 125$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3}) + 216 (- 6 x^{2} \cdot 25) + 216 (60 x \cdot 125) + 216 (- 1000 \cdot 125)}{125}}}{6} + 10)$

Step 5.3.3.1.10.1.5

Factor $216$ out of $216 (x^{3}) + 216 (- 6 x^{2} \cdot 25)$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3} - 6 x^{2} \cdot 25) + 216 (60 x \cdot 125) + 216 (- 1000 \cdot 125)}{125}}}{6} + 10)$

Step 5.3.3.1.10.1.6

Factor $216$ out of $216 (x^{3} - 6 x^{2} \cdot 25) + 216 (60 x \cdot 125)$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3} - 6 x^{2} \cdot 25 + 60 x \cdot 125) + 216 (- 1000 \cdot 125)}{125}}}{6} + 10)$

Step 5.3.3.1.10.1.7

Factor $216$ out of $216 (x^{3} - 6 x^{2} \cdot 25 + 60 x \cdot 125) + 216 (- 1000 \cdot 125)$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3} - 6 x^{2} \cdot 25 + 60 x \cdot 125 - 1000 \cdot 125)}{125}}}{6} + 10)$

Step 5.3.3.1.10.2

Multiply $25$ by $- 6$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3} - 150 x^{2} + 60 x \cdot 125 - 1000 \cdot 125)}{125}}}{6} + 10)$

Step 5.3.3.1.10.3

Multiply $125$ by $60$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3} - 150 x^{2} + 7500 x - 1000 \cdot 125)}{125}}}{6} + 10)$

Step 5.3.3.1.10.4

Multiply $- 1000$ by $125$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 (x^{3} - 150 x^{2} + 7500 x - 125000)}{125}}}{6} + 10)$

Step 5.3.3.1.10.5

Rewrite $x^{3} - 150 x^{2} + 7500 x - 125000$ in a factored form.

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Step 5.3.3.1.10.5.1

Factor $x^{3} - 150 x^{2} + 7500 x - 125000$ using the rational roots test.

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Step 5.3.3.1.10.5.1.1

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

$p = \pm 1, \pm 125000, \pm 2, \pm 62500, \pm 4, \pm 31250, \pm 5, \pm 25000, \pm 8, \pm 15625, \pm 10, \pm 12500, \pm 20, \pm 6250, \pm 25, \pm 5000, \pm 40, \pm 3125, \pm 50, \pm 2500, \pm 100, \pm 1250, \pm 125, \pm 1000, \pm 200, \pm 625, \pm 250, \pm 500$

$q = \pm 1$

Step 5.3.3.1.10.5.1.2

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

$\pm 1, \pm 125000, \pm 2, \pm 62500, \pm 4, \pm 31250, \pm 5, \pm 25000, \pm 8, \pm 15625, \pm 10, \pm 12500, \pm 20, \pm 6250, \pm 25, \pm 5000, \pm 40, \pm 3125, \pm 50, \pm 2500, \pm 100, \pm 1250, \pm 125, \pm 1000, \pm 200, \pm 625, \pm 250, \pm 500$

Step 5.3.3.1.10.5.1.3

Substitute $50$ and simplify the expression. In this case, the expression is equal to $0$ so $50$ is a root of the polynomial.

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Step 5.3.3.1.10.5.1.3.1

Substitute $50$ into the polynomial.

$50^{3} - 150 \cdot 50^{2} + 7500 \cdot 50 - 125000$

Step 5.3.3.1.10.5.1.3.2

Raise $50$ to the power of $3$ .

$125000 - 150 \cdot 50^{2} + 7500 \cdot 50 - 125000$

Step 5.3.3.1.10.5.1.3.3

Raise $50$ to the power of $2$ .

$125000 - 150 \cdot 2500 + 7500 \cdot 50 - 125000$

Step 5.3.3.1.10.5.1.3.4

Multiply $- 150$ by $2500$ .

$125000 - 375000 + 7500 \cdot 50 - 125000$

Step 5.3.3.1.10.5.1.3.5

Subtract $375000$ from $125000$ .

$- 250000 + 7500 \cdot 50 - 125000$

Step 5.3.3.1.10.5.1.3.6

Multiply $7500$ by $50$ .

$- 250000 + 375000 - 125000$

Step 5.3.3.1.10.5.1.3.7

Add $- 250000$ and $375000$ .

$125000 - 125000$

Step 5.3.3.1.10.5.1.3.8

Subtract $125000$ from $125000$ .

$0$

Step 5.3.3.1.10.5.1.4

Since $50$ is a known root, divide the polynomial by $x - 50$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 150 x^{2} + 7500 x - 125000}{x - 50}$

Step 5.3.3.1.10.5.1.5

Divide $x^{3} - 150 x^{2} + 7500 x - 125000$ by $x - 50$ .

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Step 5.3.3.1.10.5.1.5.1

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

$x$

$50$

$x^{3}$

$150 x^{2}$

$7500 x$

$125000$

Step 5.3.3.1.10.5.1.5.2

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

					$x^{2}$
	$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$

Step 5.3.3.1.10.5.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			+	$x^{3}$	-	$50 x^{2}$

Step 5.3.3.1.10.5.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 50 x^{2}$

				$x^{2}$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$

Step 5.3.3.1.10.5.1.5.5

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

				$x^{2}$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$

Step 5.3.3.1.10.5.1.5.6

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

				$x^{2}$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$

Step 5.3.3.1.10.5.1.5.7

Divide the highest order term in the dividend $- 100 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$100 x$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$

Step 5.3.3.1.10.5.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$100 x$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					-	$100 x^{2}$	+	$5000 x$

Step 5.3.3.1.10.5.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 100 x^{2} + 5000 x$

				$x^{2}$	-	$100 x$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$

Step 5.3.3.1.10.5.1.5.10

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

				$x^{2}$	-	$100 x$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$
							+	$2500 x$

Step 5.3.3.1.10.5.1.5.11

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

				$x^{2}$	-	$100 x$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$
							+	$2500 x$	-	$125000$

Step 5.3.3.1.10.5.1.5.12

Divide the highest order term in the dividend $2500 x$ by the highest order term in divisor $x$ .

				$x^{2}$	-	$100 x$	+	$2500$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$
							+	$2500 x$	-	$125000$

Step 5.3.3.1.10.5.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	-	$100 x$	+	$2500$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$
							+	$2500 x$	-	$125000$
							+	$2500 x$	-	$125000$

Step 5.3.3.1.10.5.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2500 x - 125000$

				$x^{2}$	-	$100 x$	+	$2500$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$
							+	$2500 x$	-	$125000$
							-	$2500 x$	+	$125000$

Step 5.3.3.1.10.5.1.5.15

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

				$x^{2}$	-	$100 x$	+	$2500$
$x$	-	$50$		$x^{3}$	-	$150 x^{2}$	+	$7500 x$	-	$125000$
			-	$x^{3}$	+	$50 x^{2}$
					-	$100 x^{2}$	+	$7500 x$
					+	$100 x^{2}$	-	$5000 x$
							+	$2500 x$	-	$125000$
							-	$2500 x$	+	$125000$
										$0$

Step 5.3.3.1.10.5.1.5.16

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

$x^{2} - 100 x + 2500$

Step 5.3.3.1.10.5.1.6

Write $x^{3} - 150 x^{2} + 7500 x - 125000$ as a set of factors.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 ((x - 50) (x^{2} - 100 x + 2500))}{125}}}{6} + 10)$

Step 5.3.3.1.10.5.2

Factor using the perfect square rule.

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Step 5.3.3.1.10.5.2.1

Rewrite $2500$ as $50^{2}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 ((x - 50) (x^{2} - 100 x + 50^{2}))}{125}}}{6} + 10)$

Step 5.3.3.1.10.5.2.2

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

$100 x = 2 \cdot x \cdot 50$

Step 5.3.3.1.10.5.2.3

Rewrite the polynomial.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 ((x - 50) (x^{2} - 2 \cdot x \cdot 50 + 50^{2}))}{125}}}{6} + 10)$

Step 5.3.3.1.10.5.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 50$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 ((x - 50) {(x - 50)}^{2})}{125}}}{6} + 10)$

Step 5.3.3.1.10.5.3

Combine like factors.

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Step 5.3.3.1.10.5.3.1

Raise $x - 50$ to the power of $1$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 ((x - 50) {(x - 50)}^{2})}{125}}}{6} + 10)$

Step 5.3.3.1.10.5.3.2

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

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 {(x - 50)}^{1 + 2}}{125}}}{6} + 10)$

Step 5.3.3.1.10.5.3.3

Add $1$ and $2$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{216 {(x - 50)}^{3}}{125}}}{6} + 10)$

Step 5.3.3.1.11

Rewrite $216 {(x - 50)}^{3}$ as ${(6 (x - 50))}^{3}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{{(6 (x - 50))}^{3}}{125}}}{6} + 10)$

Step 5.3.3.1.12

Rewrite $125$ as $5^{3}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{\frac{{(6 (x - 50))}^{3}}{5^{3}}}}{6} + 10)$

Step 5.3.3.1.13

Rewrite $\frac{{(6 (x - 50))}^{3}}{5^{3}}$ as ${(\frac{6 (x - 50)}{5})}^{3}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\sqrt[3]{{(\frac{6 (x - 50)}{5})}^{3}}}{6} + 10)$

Step 5.3.3.1.14

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

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{\frac{6 (x - 50)}{5}}{6} + 10)$

Step 5.3.3.2

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{6 (x - 50)}{5} \cdot \frac{1}{6} + 10)$

Step 5.3.3.3

Cancel the common factor of $6$ .

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Step 5.3.3.3.1

Cancel the common factor.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{6 (x - 50)}{5} \cdot \frac{1}{6} + 10)$

Step 5.3.3.3.2

Rewrite the expression.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x - 50}{5} + 10)$

Step 5.3.4

To write $10$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x - 50}{5} + 10 \cdot \frac{5}{5})$

Step 5.3.5

Simplify terms.

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Step 5.3.5.1

Combine $10$ and $\frac{5}{5}$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x - 50}{5} + \frac{10 \cdot 5}{5})$

Step 5.3.5.2

Combine the numerators over the common denominator.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x - 50 + 10 \cdot 5}{5})$

Step 5.3.6

Simplify the numerator.

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Step 5.3.6.1

Multiply $10$ by $5$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x - 50 + 50}{5})$

Step 5.3.6.2

Add $- 50$ and $50$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x + 0}{5})$

Step 5.3.6.3

Add $x$ and $0$ .

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x}{5})$

Step 5.3.7

Cancel the common factor of $5$ .

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Step 5.3.7.1

Cancel the common factor.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = 5 (\frac{x}{5})$

Step 5.3.7.2

Rewrite the expression.

$f (\frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000$ is the inverse of $f (x) = 5 (\frac{\sqrt[3]{x}}{6} + 10)$ .

$f^{- 1} (x) = \frac{216 x^{3}}{125} - \frac{1296 x^{2}}{5} + 12960 x - 216000$