Precalculus Examples

$\frac{{(u^{- 1} v^{2})}^{3}}{{(u^{3} v^{- 2})}^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $u^{- 1} v^{2}$ .

$\frac{{(u^{- 1})}^{3} {(v^{2})}^{3}}{{(u^{3} v^{- 2})}^{2}}$

Step 1.2

Multiply the exponents in ${(u^{- 1})}^{3}$ .

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Step 1.2.1

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

$\frac{u^{- 1 \cdot 3} {(v^{2})}^{3}}{{(u^{3} v^{- 2})}^{2}}$

Step 1.2.2

Multiply $- 1$ by $3$ .

$\frac{u^{- 3} {(v^{2})}^{3}}{{(u^{3} v^{- 2})}^{2}}$

Step 1.3

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

$\frac{\frac{1}{u^{3}} {(v^{2})}^{3}}{{(u^{3} v^{- 2})}^{2}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{\frac{1}{u^{3}} v^{2 \cdot 3}}{{(u^{3} v^{- 2})}^{2}}$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{\frac{1}{u^{3}} v^{6}}{{(u^{3} v^{- 2})}^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $u^{3} v^{- 2}$ .

$\frac{\frac{1}{u^{3}} v^{6}}{{(u^{3})}^{2} {(v^{- 2})}^{2}}$

Step 2.2

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

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Step 2.2.1

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

$\frac{\frac{1}{u^{3}} v^{6}}{u^{3 \cdot 2} {(v^{- 2})}^{2}}$

Step 2.2.2

Multiply $3$ by $2$ .

$\frac{\frac{1}{u^{3}} v^{6}}{u^{6} {(v^{- 2})}^{2}}$

Step 2.3

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

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Step 2.3.1

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

$\frac{\frac{1}{u^{3}} v^{6}}{u^{6} v^{- 2 \cdot 2}}$

Step 2.3.2

Multiply $- 2$ by $2$ .

$\frac{\frac{1}{u^{3}} v^{6}}{u^{6} v^{- 4}}$

Step 2.4

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

$\frac{\frac{1}{u^{3}} v^{6}}{u^{6} \frac{1}{v^{4}}}$

Step 3

Combine fractions.

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Step 3.1

Combine $\frac{1}{u^{3}}$ and $v^{6}$ .

$\frac{\frac{v^{6}}{u^{3}}}{u^{6} \frac{1}{v^{4}}}$

Step 3.2

Combine $u^{6}$ and $\frac{1}{v^{4}}$ .

$\frac{\frac{v^{6}}{u^{3}}}{\frac{u^{6}}{v^{4}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{v^{6}}{u^{3}} \cdot \frac{v^{4}}{u^{6}}$

Step 5

Combine.

$\frac{v^{6} v^{4}}{u^{3} u^{6}}$

Step 6

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

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Step 6.1

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

$\frac{v^{6 + 4}}{u^{3} u^{6}}$

Step 6.2

Add $6$ and $4$ .

$\frac{v^{10}}{u^{3} u^{6}}$

Step 7

Multiply $u^{3}$ by $u^{6}$ 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.

$\frac{v^{10}}{u^{3 + 6}}$

Step 7.2

Add $3$ and $6$ .

$\frac{v^{10}}{u^{9}}$