Precalculus Examples

$\frac{{(x^{- 3} y^{2})}^{- 4}}{{(y^{6} x^{- 4})}^{- 2}}$

Step 1

Simplify the expression.

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Step 1.1

Move ${(x^{- 3} y^{2})}^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(y^{6} x^{- 4})}^{- 2} {(x^{- 3} y^{2})}^{4}}$

Step 1.2

Move ${(y^{6} x^{- 4})}^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{{(y^{6} x^{- 4})}^{2}}{{(x^{- 3} y^{2})}^{4}}$

Step 2

Simplify the numerator.

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Step 2.1

Apply the product rule to $y^{6} x^{- 4}$ .

$\frac{{(y^{6})}^{2} {(x^{- 4})}^{2}}{{(x^{- 3} y^{2})}^{4}}$

Step 2.2

Multiply the exponents in ${(y^{6})}^{2}$ .

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Step 2.2.1

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

$\frac{y^{6 \cdot 2} {(x^{- 4})}^{2}}{{(x^{- 3} y^{2})}^{4}}$

Step 2.2.2

Multiply $6$ by $2$ .

$\frac{y^{12} {(x^{- 4})}^{2}}{{(x^{- 3} y^{2})}^{4}}$

Step 2.3

Multiply the exponents in ${(x^{- 4})}^{2}$ .

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Step 2.3.1

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

$\frac{y^{12} x^{- 4 \cdot 2}}{{(x^{- 3} y^{2})}^{4}}$

Step 2.3.2

Multiply $- 4$ by $2$ .

$\frac{y^{12} x^{- 8}}{{(x^{- 3} y^{2})}^{4}}$

Step 2.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{y^{12} \frac{1}{x^{8}}}{{(x^{- 3} y^{2})}^{4}}$

Step 3

Simplify the denominator.

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Step 3.1

Apply the product rule to $x^{- 3} y^{2}$ .

$\frac{y^{12} \frac{1}{x^{8}}}{{(x^{- 3})}^{4} {(y^{2})}^{4}}$

Step 3.2

Multiply the exponents in ${(x^{- 3})}^{4}$ .

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Step 3.2.1

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

$\frac{y^{12} \frac{1}{x^{8}}}{x^{- 3 \cdot 4} {(y^{2})}^{4}}$

Step 3.2.2

Multiply $- 3$ by $4$ .

$\frac{y^{12} \frac{1}{x^{8}}}{x^{- 12} {(y^{2})}^{4}}$

Step 3.3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{y^{12} \frac{1}{x^{8}}}{\frac{1}{x^{12}} {(y^{2})}^{4}}$

Step 3.4

Multiply the exponents in ${(y^{2})}^{4}$ .

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Step 3.4.1

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

$\frac{y^{12} \frac{1}{x^{8}}}{\frac{1}{x^{12}} y^{2 \cdot 4}}$

Step 3.4.2

Multiply $2$ by $4$ .

$\frac{y^{12} \frac{1}{x^{8}}}{\frac{1}{x^{12}} y^{8}}$

Step 4

Combine fractions.

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Step 4.1

Combine $y^{12}$ and $\frac{1}{x^{8}}$ .

$\frac{\frac{y^{12}}{x^{8}}}{\frac{1}{x^{12}} y^{8}}$

Step 4.2

Combine $\frac{1}{x^{12}}$ and $y^{8}$ .

$\frac{\frac{y^{12}}{x^{8}}}{\frac{y^{8}}{x^{12}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{12}}{x^{8}} \cdot \frac{x^{12}}{y^{8}}$

Step 6

Combine.

$\frac{y^{12} x^{12}}{x^{8} y^{8}}$

Step 7

Cancel the common factor of $y^{12}$ and $y^{8}$ .

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Step 7.1

Factor $y^{8}$ out of $y^{12} x^{12}$ .

$\frac{y^{8} (y^{4} x^{12})}{x^{8} y^{8}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $y^{8}$ out of $x^{8} y^{8}$ .

$\frac{y^{8} (y^{4} x^{12})}{y^{8} x^{8}}$

Step 7.2.2

Cancel the common factor.

$\frac{y^{8} (y^{4} x^{12})}{y^{8} x^{8}}$

Step 7.2.3

Rewrite the expression.

$\frac{y^{4} x^{12}}{x^{8}}$

Step 8

Cancel the common factor of $x^{12}$ and $x^{8}$ .

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Step 8.1

Factor $x^{8}$ out of $y^{4} x^{12}$ .

$\frac{x^{8} (y^{4} x^{4})}{x^{8}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Multiply by $1$ .

$\frac{x^{8} (y^{4} x^{4})}{x^{8} \cdot 1}$

Step 8.2.2

Cancel the common factor.

$\frac{x^{8} (y^{4} x^{4})}{x^{8} \cdot 1}$

Step 8.2.3

Rewrite the expression.

$\frac{y^{4} x^{4}}{1}$

Step 8.2.4

Divide $y^{4} x^{4}$ by $1$ .

$y^{4} x^{4}$