Algebra Examples

$x^{3} y = - 9$

Step 1

Divide each term in $x^{3} y = - 9$ by $x^{3}$ and simplify.

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Step 1.1

Divide each term in $x^{3} y = - 9$ by $x^{3}$ .

$\frac{x^{3} y}{x^{3}} = \frac{- 9}{x^{3}}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $x^{3}$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{x^{3} y}{x^{3}} = \frac{- 9}{x^{3}}$

Step 1.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 9}{x^{3}}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$y = - \frac{9}{x^{3}}$

Step 2

Interchange the variables.

$x = - \frac{9}{y^{3}}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{9}{y^{3}} = x$ .

$- \frac{9}{y^{3}} = x$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$y^{3}, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$y^{3}$

Step 3.3

Multiply each term in $- \frac{9}{y^{3}} = x$ by $y^{3}$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $- \frac{9}{y^{3}} = x$ by $y^{3}$ .

$- \frac{9}{y^{3}} y^{3} = x y^{3}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $y^{3}$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{9}{y^{3}}$ into the numerator.

$\frac{- 9}{y^{3}} y^{3} = x y^{3}$

Step 3.3.2.1.2

Cancel the common factor.

$\frac{- 9}{y^{3}} y^{3} = x y^{3}$

Step 3.3.2.1.3

Rewrite the expression.

$- 9 = x y^{3}$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $x y^{3} = - 9$ .

$x y^{3} = - 9$

Step 3.4.2

Divide each term in $x y^{3} = - 9$ by $x$ and simplify.

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Step 3.4.2.1

Divide each term in $x y^{3} = - 9$ by $x$ .

$\frac{x y^{3}}{x} = \frac{- 9}{x}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $x$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{x y^{3}}{x} = \frac{- 9}{x}$

Step 3.4.2.2.1.2

Divide $y^{3}$ by $1$ .

$y^{3} = \frac{- 9}{x}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Move the negative in front of the fraction.

$y^{3} = - \frac{9}{x}$

Step 3.4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \sqrt[3]{- \frac{9}{x}}$

Step 3.4.4

Simplify $\sqrt[3]{- \frac{9}{x}}$ .

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Step 3.4.4.1

Rewrite $- \frac{9}{x}$ as ${({(- 1)}^{3})}^{3} \frac{9}{x}$ .

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Step 3.4.4.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{(- 1)}^{3} \frac{9}{x}}$

Step 3.4.4.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$y = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{9}{x}}$

Step 3.4.4.2

Pull terms out from under the radical.

$y = {(- 1)}^{3} \sqrt[3]{\frac{9}{x}}$

Step 3.4.4.3

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

$y = - \sqrt[3]{\frac{9}{x}}$

Step 3.4.4.4

Rewrite $\sqrt[3]{\frac{9}{x}}$ as $\frac{\sqrt[3]{9}}{\sqrt[3]{x}}$ .

$y = - \frac{\sqrt[3]{9}}{\sqrt[3]{x}}$

Step 3.4.4.5

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = - (\frac{\sqrt[3]{9}}{\sqrt[3]{x}} \cdot \frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}})$

Step 3.4.4.6

Combine and simplify the denominator.

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Step 3.4.4.6.1

Multiply $\frac{\sqrt[3]{9}}{\sqrt[3]{x}}$ by $\frac{{\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{2}}$ .

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{\sqrt[3]{x} {\sqrt[3]{x}}^{2}}$

Step 3.4.4.6.2

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

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1} {\sqrt[3]{x}}^{2}}$

Step 3.4.4.6.3

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

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{1 + 2}}$

Step 3.4.4.6.4

Add $1$ and $2$ .

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{{\sqrt[3]{x}}^{3}}$

Step 3.4.4.6.5

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

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Step 3.4.4.6.5.1

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

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{{(x^{\frac{1}{3}})}^{3}}$

Step 3.4.4.6.5.2

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

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{x^{\frac{1}{3} \cdot 3}}$

Step 3.4.4.6.5.3

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

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 3.4.4.6.5.4

Cancel the common factor of $3$ .

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Step 3.4.4.6.5.4.1

Cancel the common factor.

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{x^{\frac{3}{3}}}$

Step 3.4.4.6.5.4.2

Rewrite the expression.

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{x^{1}}$

Step 3.4.4.6.5.5

Simplify.

$y = - \frac{\sqrt[3]{9} {\sqrt[3]{x}}^{2}}{x}$

Step 3.4.4.7

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

$y = - \frac{\sqrt[3]{9} \sqrt[3]{x^{2}}}{x}$

Step 3.4.4.8

Combine using the product rule for radicals.

$y = - \frac{\sqrt[3]{9 x^{2}}}{x}$

Step 4

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

$f^{- 1} (x) = - \frac{\sqrt[3]{9 x^{2}}}{x}$

Step 5

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

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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 $f^{- 1} (- \frac{9}{x^{3}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - \frac{\sqrt[3]{9 {(- \frac{9}{x^{3}})}^{2}}}{- \frac{9}{x^{3}}}$

Step 5.2.3

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (- \frac{9}{x^{3}}) = - (\sqrt[3]{9 {(- \frac{9}{x^{3}})}^{2}} (- \frac{x^{3}}{9}))$

Step 5.2.4

Rewrite using the commutative property of multiplication.

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{9 {(- \frac{9}{x^{3}})}^{2}} \frac{x^{3}}{9})$

Step 5.2.5

Apply the product rule to $- \frac{9}{x^{3}}$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{9 {(- 1)}^{2} {(\frac{9}{x^{3}})}^{2}} \frac{x^{3}}{9})$

Step 5.2.6

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

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

Step 5.2.7

Apply the product rule to $\frac{9}{x^{3}}$ .

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

Step 5.2.8

Raise $9$ to the power of $2$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{9 \cdot (1 (\frac{81}{{(x^{3})}^{2}}))} \frac{x^{3}}{9})$

Step 5.2.9

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 5.2.9.1

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

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{9 \cdot (1 (\frac{81}{x^{3 \cdot 2}}))} \frac{x^{3}}{9})$

Step 5.2.9.2

Multiply $3$ by $2$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{9 \cdot (1 (\frac{81}{x^{6}}))} \frac{x^{3}}{9})$

Step 5.2.10

Combine exponents.

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Step 5.2.10.1

Multiply $9$ by $1$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{9 (\frac{81}{x^{6}})} \frac{x^{3}}{9})$

Step 5.2.10.2

Combine $9$ and $\frac{81}{x^{6}}$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{\frac{9 \cdot 81}{x^{6}}} \frac{x^{3}}{9})$

Step 5.2.10.3

Multiply $9$ by $81$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{\frac{729}{x^{6}}} \frac{x^{3}}{9})$

Step 5.2.11

Rewrite $729$ as $9^{3}$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{\frac{9^{3}}{x^{6}}} \frac{x^{3}}{9})$

Step 5.2.12

Rewrite $x^{6}$ as ${(x^{2})}^{3}$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{\frac{9^{3}}{{(x^{2})}^{3}}} \frac{x^{3}}{9})$

Step 5.2.13

Rewrite $\frac{9^{3}}{{(x^{2})}^{3}}$ as ${(\frac{9}{x^{2}})}^{3}$ .

$f^{- 1} (- \frac{9}{x^{3}}) = - (- \sqrt[3]{{(\frac{9}{x^{2}})}^{3}} \frac{x^{3}}{9})$

Step 5.2.14

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

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

Step 5.2.15

Cancel the common factor of $9$ .

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Step 5.2.15.1

Move the leading negative in $- \frac{9}{x^{2}}$ into the numerator.

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

Step 5.2.15.2

Factor $9$ out of $- 9$ .

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

Step 5.2.15.3

Cancel the common factor.

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

Step 5.2.15.4

Rewrite the expression.

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

Step 5.2.16

Cancel the common factor of $x^{2}$ .

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Step 5.2.16.1

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

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

Step 5.2.16.2

Cancel the common factor.

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

Step 5.2.16.3

Rewrite the expression.

$f^{- 1} (- \frac{9}{x^{3}}) = x$

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 $f (- \frac{\sqrt[3]{9 x^{2}}}{x})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{{(- \frac{\sqrt[3]{9 x^{2}}}{x})}^{3}}$

Step 5.3.3

Simplify the denominator.

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Step 5.3.3.1

Apply the product rule to $- \frac{\sqrt[3]{9 x^{2}}}{x}$ .

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{{(- 1)}^{3} {(\frac{\sqrt[3]{9 x^{2}}}{x})}^{3}}$

Step 5.3.3.2

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

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- {(\frac{\sqrt[3]{9 x^{2}}}{x})}^{3}}$

Step 5.3.3.3

Apply the product rule to $\frac{\sqrt[3]{9 x^{2}}}{x}$ .

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{{\sqrt[3]{9 x^{2}}}^{3}}{x^{3}}}$

Step 5.3.3.4

Rewrite ${\sqrt[3]{9 x^{2}}}^{3}$ as $9 x^{2}$ .

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Step 5.3.3.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{9 x^{2}}$ as ${(9 x^{2})}^{\frac{1}{3}}$ .

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{{({(9 x^{2})}^{\frac{1}{3}})}^{3}}{x^{3}}}$

Step 5.3.3.4.2

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

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

Step 5.3.3.4.3

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

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{{(9 x^{2})}^{\frac{3}{3}}}{x^{3}}}$

Step 5.3.3.4.4

Cancel the common factor of $3$ .

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Step 5.3.3.4.4.1

Cancel the common factor.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{{(9 x^{2})}^{\frac{3}{3}}}{x^{3}}}$

Step 5.3.3.4.4.2

Rewrite the expression.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{9 x^{2}}{x^{3}}}$

Step 5.3.3.4.5

Simplify.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{9 x^{2}}{x^{3}}}$

Step 5.3.3.5

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 5.3.3.5.1

Factor $x^{2}$ out of $9 x^{2}$ .

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{x^{2} \cdot 9}{x^{3}}}$

Step 5.3.3.5.2

Cancel the common factors.

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Step 5.3.3.5.2.1

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

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{x^{2} \cdot 9}{x^{2} x}}$

Step 5.3.3.5.2.2

Cancel the common factor.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{x^{2} \cdot 9}{x^{2} x}}$

Step 5.3.3.5.2.3

Rewrite the expression.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - \frac{9}{- \frac{9}{x}}$

Step 5.3.4

Multiply the numerator by the reciprocal of the denominator.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - (9 (- \frac{x}{9}))$

Step 5.3.5

Cancel the common factor of $9$ .

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Step 5.3.5.1

Move the leading negative in $- \frac{x}{9}$ into the numerator.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - (9 (\frac{- x}{9}))$

Step 5.3.5.2

Cancel the common factor.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = - (9 (\frac{- x}{9}))$

Step 5.3.5.3

Rewrite the expression.

$f (- \frac{\sqrt[3]{9 x^{2}}}{x}) = x$

Step 5.4

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

$f^{- 1} (x) = - \frac{\sqrt[3]{9 x^{2}}}{x}$