Find the Inverse cube root of 27x-81-5

$\sqrt[3]{27 x - 81} - 5$

Interchange the variables.

$x = \sqrt[3]{27 y - 81} - 5$

Solve for $y$ .

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Rewrite the equation as $\sqrt[3]{27 y - 81} - 5 = x$ .

$\sqrt[3]{27 y - 81} - 5 = x$

Add $5$ to both sides of the equation.

$\sqrt[3]{27 y - 81} = x + 5$

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

${\sqrt[3]{27 y - 81}}^{3} = {(x + 5)}^{3}$

Simplify each side of the equation.

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Multiply the exponents in ${({(27 y - 81)}^{\frac{1}{3}})}^{3}$ .

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

${(27 y - 81)}^{\frac{1}{3} \cdot 3} = {(x + 5)}^{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

${(27 y - 81)}^{\frac{1}{3} \cdot 3} = {(x + 5)}^{3}$

Rewrite the expression.

${(27 y - 81)}^{1} = {(x + 5)}^{3}$

Simplify.

$27 y - 81 = {(x + 5)}^{3}$

Solve for $y$ .

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Simplify ${(x + 5)}^{3}$ .

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Use the Binomial Theorem.

$27 y - 81 = x^{3} + 3 x^{2} \cdot 5 + 3 x \cdot 5^{2} + 5^{3}$

Simplify each term.

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Multiply $5$ by $3$ .

$27 y - 81 = x^{3} + 15 x^{2} + 3 x \cdot 5^{2} + 5^{3}$

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

$27 y - 81 = x^{3} + 15 x^{2} + 3 x \cdot 25 + 5^{3}$

Multiply $25$ by $3$ .

$27 y - 81 = x^{3} + 15 x^{2} + 75 x + 5^{3}$

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

$27 y - 81 = x^{3} + 15 x^{2} + 75 x + 125$

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

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Add $81$ to both sides of the equation.

$27 y = x^{3} + 15 x^{2} + 75 x + 125 + 81$

Add $125$ and $81$ .

$27 y = x^{3} + 15 x^{2} + 75 x + 206$

Divide each term by $27$ and simplify.

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Divide each term in $27 y = x^{3} + 15 x^{2} + 75 x + 206$ by $27$ .

$\frac{27 y}{27} = \frac{x^{3}}{27} + \frac{15 x^{2}}{27} + \frac{75 x}{27} + \frac{206}{27}$

Cancel the common factor of $27$ .

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Cancel the common factor.

$\frac{27 y}{27} = \frac{x^{3}}{27} + \frac{15 x^{2}}{27} + \frac{75 x}{27} + \frac{206}{27}$

Divide $y$ by $1$ .

$y = \frac{x^{3}}{27} + \frac{15 x^{2}}{27} + \frac{75 x}{27} + \frac{206}{27}$

Simplify each term.

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Cancel the common factor of $15$ and $27$ .

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Factor $3$ out of $15 x^{2}$ .

$y = \frac{x^{3}}{27} + \frac{3 (5 x^{2})}{27} + \frac{75 x}{27} + \frac{206}{27}$

Cancel the common factors.

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Factor $3$ out of $27$ .

$y = \frac{x^{3}}{27} + \frac{3 (5 x^{2})}{3 (9)} + \frac{75 x}{27} + \frac{206}{27}$

Cancel the common factor.

$y = \frac{x^{3}}{27} + \frac{3 (5 x^{2})}{3 \cdot 9} + \frac{75 x}{27} + \frac{206}{27}$

Rewrite the expression.

$y = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{75 x}{27} + \frac{206}{27}$

Cancel the common factor of $75$ and $27$ .

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Factor $3$ out of $75 x$ .

$y = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{3 (25 x)}{27} + \frac{206}{27}$

Cancel the common factors.

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Factor $3$ out of $27$ .

$y = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{3 (25 x)}{3 (9)} + \frac{206}{27}$

Cancel the common factor.

$y = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{3 (25 x)}{3 \cdot 9} + \frac{206}{27}$

Rewrite the expression.

$y = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{25 x}{9} + \frac{206}{27}$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

$f^{- 1} (x) = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{25 x}{9} + \frac{206}{27}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(\sqrt[3]{27 x - 81} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Simplify each term.

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Simplify the numerator.

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Factor $27$ out of $27 x - 81$ .

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Factor $27$ out of $27 x$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(\sqrt[3]{27 (x) - 81} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Factor $27$ out of $- 81$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(\sqrt[3]{27 x + 27 \cdot - 3} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Factor $27$ out of $27 x + 27 \cdot - 3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(\sqrt[3]{27 (x - 3)} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(\sqrt[3]{3^{3} (x - 3)} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Pull terms out from under the radical.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Simplify the numerator.

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Factor $27$ out of $27 x - 81$ .

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Factor $27$ out of $27 x$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 (x) - 81} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Factor $27$ out of $- 81$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 x + 27 \cdot - 3} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Factor $27$ out of $27 x + 27 \cdot - 3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{27 (x - 3)} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(\sqrt[3]{3^{3} (x - 3)} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Pull terms out from under the radical.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x - 81} - 5)}{9} + \frac{206}{27}$

Simplify the numerator.

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Factor $27$ out of $27 x - 81$ .

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Factor $27$ out of $27 x$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 (x) - 81} - 5)}{9} + \frac{206}{27}$

Factor $27$ out of $- 81$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 x + 27 \cdot - 3} - 5)}{9} + \frac{206}{27}$

Factor $27$ out of $27 x + 27 \cdot - 3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{27 (x - 3)} - 5)}{9} + \frac{206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} + \frac{25 (\sqrt[3]{3^{3} (x - 3)} - 5)}{9} + \frac{206}{27}$

Pull terms out from under the radical.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

To write $\frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9} \cdot \frac{3}{3} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

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

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Multiply $\frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2}}{9}$ and $\frac{3}{3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2} \cdot 3}{9 \cdot 3} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

Multiply $9$ by $3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3}}{27} + \frac{5 {(3 \sqrt[3]{x - 3} - 5)}^{2} \cdot 3}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

Combine the numerators over the common denominator.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{3} + 5 {(3 \sqrt[3]{x - 3} - 5)}^{2} \cdot 3}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

Simplify the numerator.

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

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} - 5) + 5 {(3 \sqrt[3]{x - 3} - 5)}^{2} \cdot 3}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

Factor ${(3 \sqrt[3]{x - 3} - 5)}^{2}$ out of $5 {(3 \sqrt[3]{x - 3} - 5)}^{2} \cdot 3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} - 5) + {(3 \sqrt[3]{x - 3} - 5)}^{2} (5 \cdot 3)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} - 5 + 5 \cdot 3)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

Multiply $5$ by $3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} - 5 + 15)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

Add $- 5$ and $15$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} + \frac{206}{27}$

To write $\frac{25 (3 \sqrt[3]{x - 3} - 5)}{9}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5)}{9} \cdot \frac{3}{3} + \frac{206}{27}$

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

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Multiply $\frac{25 (3 \sqrt[3]{x - 3} - 5)}{9}$ and $\frac{3}{3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5) \cdot 3}{9 \cdot 3} + \frac{206}{27}$

Multiply $9$ by $3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10)}{27} + \frac{25 (3 \sqrt[3]{x - 3} - 5) \cdot 3}{27} + \frac{206}{27}$

Combine into one fraction.

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Combine the numerators over the common denominator.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10) + 25 (3 \sqrt[3]{x - 3} - 5) \cdot 3}{27} + \frac{206}{27}$

Reorder terms.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5))}{27} + \frac{206}{27}$

Combine the numerators over the common denominator.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{{(3 \sqrt[3]{x - 3} - 5)}^{2} (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Simplify the numerator.

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Rewrite ${(3 \sqrt[3]{x - 3} - 5)}^{2}$ as $(3 \sqrt[3]{x - 3} - 5) (3 \sqrt[3]{x - 3} - 5)$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(3 \sqrt[3]{x - 3} - 5) (3 \sqrt[3]{x - 3} - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Expand $(3 \sqrt[3]{x - 3} - 5) (3 \sqrt[3]{x - 3} - 5)$ using the FOIL Method.

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Apply the distributive property.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(3 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3} - 5) - 5 (3 \sqrt[3]{x - 3} - 5)) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Apply the distributive property.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(3 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3} - 5)) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Apply the distributive property.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(3 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Simplify and combine like terms.

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Simplify each term.

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

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Multiply $3$ by $3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 \sqrt[3]{x - 3} \sqrt[3]{x - 3} + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 (\sqrt[3]{x - 3} \sqrt[3]{x - 3}) + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 {\sqrt[3]{x - 3}}^{1 + 1} + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Add $1$ and $1$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 {\sqrt[3]{x - 3}}^{2} + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 \sqrt[3]{{(x - 3)}^{2}} + 3 \sqrt[3]{x - 3} \cdot - 5 - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $- 5$ by $3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 \sqrt[3]{{(x - 3)}^{2}} - 15 \sqrt[3]{x - 3} - 5 (3 \sqrt[3]{x - 3}) - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $3$ by $- 5$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 \sqrt[3]{{(x - 3)}^{2}} - 15 \sqrt[3]{x - 3} - 15 \sqrt[3]{x - 3} - 5 \cdot - 5) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $- 5$ by $- 5$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 \sqrt[3]{{(x - 3)}^{2}} - 15 \sqrt[3]{x - 3} - 15 \sqrt[3]{x - 3} + 25) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Subtract $15 \sqrt[3]{x - 3}$ from $- 15 \sqrt[3]{x - 3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{(9 \sqrt[3]{{(x - 3)}^{2}} - 30 \sqrt[3]{x - 3} + 25) (3 \sqrt[3]{x - 3} + 10) + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Expand $(9 \sqrt[3]{{(x - 3)}^{2}} - 30 \sqrt[3]{x - 3} + 25) (3 \sqrt[3]{x - 3} + 10)$ by multiplying each term in the first expression by each term in the second expression.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{9 \sqrt[3]{{(x - 3)}^{2}} (3 \sqrt[3]{x - 3}) + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Simplify each term.

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

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Multiply $3$ by $9$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 \sqrt[3]{{(x - 3)}^{2}} \sqrt[3]{x - 3} + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Combine using the product rule for radicals.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 \sqrt[3]{(x - 3) {(x - 3)}^{2}} + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $x - 3$ by ${(x - 3)}^{2}$ by adding the exponents.

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Multiply $x - 3$ by ${(x - 3)}^{2}$ .

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

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 \sqrt[3]{{(x - 3)}^{1 + 2}} + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Add $1$ and $2$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 \sqrt[3]{{(x - 3)}^{3}} + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 (x - 3) + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Apply the distributive property.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 27 \cdot - 3 + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $27$ by $- 3$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 9 \sqrt[3]{{(x - 3)}^{2}} \cdot 10 - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $10$ by $9$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $- 30 \sqrt[3]{x - 3} (3 \sqrt[3]{x - 3})$ .

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Multiply $3$ by $- 30$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 \sqrt[3]{x - 3} \sqrt[3]{x - 3} - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 (\sqrt[3]{x - 3} \sqrt[3]{x - 3}) - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 {\sqrt[3]{x - 3}}^{1 + 1} - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Add $1$ and $1$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 {\sqrt[3]{x - 3}}^{2} - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

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

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 \sqrt[3]{{(x - 3)}^{2}} - 30 \sqrt[3]{x - 3} \cdot 10 + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $10$ by $- 30$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 \sqrt[3]{{(x - 3)}^{2}} - 300 \sqrt[3]{x - 3} + 25 (3 \sqrt[3]{x - 3}) + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $3$ by $25$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 \sqrt[3]{{(x - 3)}^{2}} - 300 \sqrt[3]{x - 3} + 75 \sqrt[3]{x - 3} + 25 \cdot 10 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $25$ by $10$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 \sqrt[3]{{(x - 3)}^{2}} - 300 \sqrt[3]{x - 3} + 75 \sqrt[3]{x - 3} + 250 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Combine the opposite terms in $27 x - 81 + 90 \sqrt[3]{{(x - 3)}^{2}} - 90 \sqrt[3]{{(x - 3)}^{2}} - 300 \sqrt[3]{x - 3} + 75 \sqrt[3]{x - 3} + 250$ .

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Subtract $90 \sqrt[3]{{(x - 3)}^{2}}$ from $90 \sqrt[3]{{(x - 3)}^{2}}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 + 0 - 300 \sqrt[3]{x - 3} + 75 \sqrt[3]{x - 3} + 250 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Add $27 x - 81$ and $0$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 81 - 300 \sqrt[3]{x - 3} + 75 \sqrt[3]{x - 3} + 250 + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Add $- 81$ and $250$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 169 - 300 \sqrt[3]{x - 3} + 75 \sqrt[3]{x - 3} + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Add $- 300 \sqrt[3]{x - 3}$ and $75 \sqrt[3]{x - 3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 169 - 225 \sqrt[3]{x - 3} + 3 \cdot (25 (3 \sqrt[3]{x - 3} - 5)) + 206}{27}$

Multiply $3$ by $25$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 169 - 225 \sqrt[3]{x - 3} + 75 (3 \sqrt[3]{x - 3} - 5) + 206}{27}$

Apply the distributive property.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 169 - 225 \sqrt[3]{x - 3} + 75 (3 \sqrt[3]{x - 3}) + 75 \cdot - 5 + 206}{27}$

Multiply $3$ by $75$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 169 - 225 \sqrt[3]{x - 3} + 225 \sqrt[3]{x - 3} + 75 \cdot - 5 + 206}{27}$

Multiply $75$ by $- 5$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 169 - 225 \sqrt[3]{x - 3} + 225 \sqrt[3]{x - 3} - 375 + 206}{27}$

Subtract $375$ from $169$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 206 - 225 \sqrt[3]{x - 3} + 225 \sqrt[3]{x - 3} + 206}{27}$

Add $- 206$ and $206$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 0 - 225 \sqrt[3]{x - 3} + 225 \sqrt[3]{x - 3}}{27}$

Add $27 x$ and $0$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x - 225 \sqrt[3]{x - 3} + 225 \sqrt[3]{x - 3}}{27}$

Add $- 225 \sqrt[3]{x - 3}$ and $225 \sqrt[3]{x - 3}$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x + 0}{27}$

Add $27 x$ and $0$ .

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x}{27}$

Cancel the common factor of $27$ .

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Cancel the common factor.

$g (\sqrt[3]{27 x - 81} - 5) = \frac{27 x}{27}$

Divide $x$ by $1$ .

$g (\sqrt[3]{27 x - 81} - 5) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{25 x}{9} + \frac{206}{27}$ is the inverse of $f (x) = \sqrt[3]{27 x - 81} - 5$ .

$f^{- 1} (x) = \frac{x^{3}}{27} + \frac{5 x^{2}}{9} + \frac{25 x}{9} + \frac{206}{27}$

Find the Inverse cube root of 27x-81-5

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