Algebra Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $- 2$ by $2$ .

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

Step 2.3

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $3$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Multiply $- 2$ by $3$ .

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

Step 3.4

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

$\frac{\frac{1}{x^{4}} y^{2}}{x^{6} \frac{1}{y^{6}}}$

Step 4

Combine fractions.

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Step 4.1

Combine $\frac{1}{x^{4}}$ and $y^{2}$ .

$\frac{\frac{y^{2}}{x^{4}}}{x^{6} \frac{1}{y^{6}}}$

Step 4.2

Combine $x^{6}$ and $\frac{1}{y^{6}}$ .

$\frac{\frac{y^{2}}{x^{4}}}{\frac{x^{6}}{y^{6}}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2}}{x^{4}} \cdot \frac{y^{6}}{x^{6}}$

Step 6

Combine.

$\frac{y^{2} y^{6}}{x^{4} x^{6}}$

Step 7

Multiply $y^{2}$ by $y^{6}$ by adding the exponents.

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Step 7.1

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

$\frac{y^{2 + 6}}{x^{4} x^{6}}$

Step 7.2

Add $2$ and $6$ .

$\frac{y^{8}}{x^{4} x^{6}}$

Step 8

Multiply $x^{4}$ by $x^{6}$ by adding the exponents.

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Step 8.1

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

$\frac{y^{8}}{x^{4 + 6}}$

Step 8.2

Add $4$ and $6$ .

$\frac{y^{8}}{x^{10}}$