Precalculus Examples

$\frac{{(3 u^{2} v^{5} w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $3 u^{2} v^{5} w^{- 4}$ .

$\frac{{(3 u^{2} v^{5})}^{3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.2.1

Apply the product rule to $3 u^{2} v^{5}$ .

$\frac{{(3 u^{2})}^{3} {(v^{5})}^{3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.2.2

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

$\frac{3^{3} {(u^{2})}^{3} {(v^{5})}^{3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.3

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

$\frac{27 {(u^{2})}^{3} {(v^{5})}^{3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.4

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

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Step 1.4.1

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

$\frac{27 u^{2 \cdot 3} {(v^{5})}^{3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{27 u^{6} {(v^{5})}^{3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.5

Multiply the exponents in ${(v^{5})}^{3}$ .

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Step 1.5.1

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

$\frac{27 u^{6} v^{5 \cdot 3} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.5.2

Multiply $5$ by $3$ .

$\frac{27 u^{6} v^{15} {(w^{- 4})}^{3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.6

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

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Step 1.6.1

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

$\frac{27 u^{6} v^{15} w^{- 4 \cdot 3}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.6.2

Multiply $- 4$ by $3$ .

$\frac{27 u^{6} v^{15} w^{- 12}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.7

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

$\frac{27 u^{6} v^{15} \frac{1}{w^{12}}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.8

Combine exponents.

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Step 1.8.1

Combine $27$ and $\frac{1}{w^{12}}$ .

$\frac{u^{6} v^{15} \frac{27}{w^{12}}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.8.2

Combine $u^{6}$ and $\frac{27}{w^{12}}$ .

$\frac{v^{15} \frac{u^{6} \cdot 27}{w^{12}}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.8.3

Combine $v^{15}$ and $\frac{u^{6} \cdot 27}{w^{12}}$ .

$\frac{\frac{v^{15} (u^{6} \cdot 27)}{w^{12}}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.9

Remove unnecessary parentheses.

$\frac{\frac{v^{15} u^{6} \cdot 27}{w^{12}}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 1.10

Move $27$ to the left of $v^{15} u^{6}$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{{(2 u v^{- 3} w^{2})}^{4}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{{(2 u v^{- 3})}^{4} {(w^{2})}^{4}}$

Step 2.2

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

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{{(2 u \frac{1}{v^{3}})}^{4} {(w^{2})}^{4}}$

Step 2.3

Multiply $2 u \frac{1}{v^{3}}$ .

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Step 2.3.1

Combine $2$ and $\frac{1}{v^{3}}$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{{(u \frac{2}{v^{3}})}^{4} {(w^{2})}^{4}}$

Step 2.3.2

Combine $u$ and $\frac{2}{v^{3}}$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{{(\frac{u \cdot 2}{v^{3}})}^{4} {(w^{2})}^{4}}$

Step 2.4

Move $2$ to the left of $u$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{{(\frac{2 \cdot u}{v^{3}})}^{4} {(w^{2})}^{4}}$

Step 2.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.5.1

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

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{{(2 u)}^{4}}{{(v^{3})}^{4}} {(w^{2})}^{4}}$

Step 2.5.2

Apply the product rule to $2 u$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{2^{4} u^{4}}{{(v^{3})}^{4}} {(w^{2})}^{4}}$

Step 2.6

Raise $2$ to the power of $4$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{16 u^{4}}{{(v^{3})}^{4}} {(w^{2})}^{4}}$

Step 2.7

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

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Step 2.7.1

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

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{16 u^{4}}{v^{3 \cdot 4}} {(w^{2})}^{4}}$

Step 2.7.2

Multiply $3$ by $4$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{16 u^{4}}{v^{12}} {(w^{2})}^{4}}$

Step 2.8

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

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Step 2.8.1

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

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{16 u^{4}}{v^{12}} w^{2 \cdot 4}}$

Step 2.8.2

Multiply $2$ by $4$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{16 u^{4}}{v^{12}} w^{8}}$

Step 3

Combine $\frac{16 u^{4}}{v^{12}}$ and $w^{8}$ .

$\frac{\frac{27 v^{15} u^{6}}{w^{12}}}{\frac{16 u^{4} w^{8}}{v^{12}}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$\frac{27 v^{15} u^{6}}{w^{12}} \cdot \frac{v^{12}}{16 u^{4} w^{8}}$

Step 5

Combine.

$\frac{27 v^{15} u^{6} v^{12}}{w^{12} (16 u^{4} w^{8})}$

Step 6

Multiply $v^{15}$ by $v^{12}$ by adding the exponents.

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Step 6.1

Move $v^{12}$ .

$\frac{27 (v^{12} v^{15}) u^{6}}{w^{12} (16 u^{4} w^{8})}$

Step 6.2

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

$\frac{27 v^{12 + 15} u^{6}}{w^{12} (16 u^{4} w^{8})}$

Step 6.3

Add $12$ and $15$ .

$\frac{27 v^{27} u^{6}}{w^{12} (16 u^{4} w^{8})}$

Step 7

Multiply $w^{12}$ by $w^{8}$ by adding the exponents.

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Step 7.1

Move $w^{8}$ .

$\frac{27 v^{27} u^{6}}{w^{8} w^{12} (16 u^{4})}$

Step 7.2

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

$\frac{27 v^{27} u^{6}}{w^{8 + 12} (16 u^{4})}$

Step 7.3

Add $8$ and $12$ .

$\frac{27 v^{27} u^{6}}{w^{20} (16 u^{4})}$

Step 8

Cancel the common factor of $u^{6}$ and $u^{4}$ .

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Step 8.1

Factor $u^{4}$ out of $27 v^{27} u^{6}$ .

$\frac{u^{4} (27 v^{27} u^{2})}{w^{20} (16 u^{4})}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $u^{4}$ out of $w^{20} (16 u^{4})$ .

$\frac{u^{4} (27 v^{27} u^{2})}{u^{4} (w^{20} \cdot (16))}$

Step 8.2.2

Cancel the common factor.

$\frac{u^{4} (27 v^{27} u^{2})}{u^{4} (w^{20} \cdot (16))}$

Step 8.2.3

Rewrite the expression.

$\frac{27 v^{27} u^{2}}{w^{20} \cdot (16)}$

Step 9

Move $16$ to the left of $w^{20}$ .

$\frac{27 v^{27} u^{2}}{16 w^{20}}$