Precalculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Raise $2$ to the power of $3$ .

$8 {(u^{2})}^{3} {(v^{3})}^{3} {(3 u^{- 3} v)}^{2}$

Step 3

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

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Step 3.1

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

$8 u^{2 \cdot 3} {(v^{3})}^{3} {(3 u^{- 3} v)}^{2}$

Step 3.2

Multiply $2$ by $3$ .

$8 u^{6} {(v^{3})}^{3} {(3 u^{- 3} v)}^{2}$

Step 4

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

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Step 4.1

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

$8 u^{6} v^{3 \cdot 3} {(3 u^{- 3} v)}^{2}$

Step 4.2

Multiply $3$ by $3$ .

$8 u^{6} v^{9} {(3 u^{- 3} v)}^{2}$

Step 5

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

$8 u^{6} v^{9} {(3 \frac{1}{u^{3}} v)}^{2}$

Step 6

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

$8 u^{6} v^{9} {(\frac{3}{u^{3}} v)}^{2}$

Step 7

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

$8 u^{6} v^{9} {(\frac{3 v}{u^{3}})}^{2}$

Step 8

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

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Step 8.1

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

$8 u^{6} v^{9} \frac{{(3 v)}^{2}}{{(u^{3})}^{2}}$

Step 8.2

Apply the product rule to $3 v$ .

$8 u^{6} v^{9} \frac{3^{2} v^{2}}{{(u^{3})}^{2}}$

Step 9

Simplify the expression.

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Step 9.1

Raise $3$ to the power of $2$ .

$8 u^{6} v^{9} \frac{9 v^{2}}{{(u^{3})}^{2}}$

Step 9.2

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

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Step 9.2.1

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

$8 u^{6} v^{9} \frac{9 v^{2}}{u^{3 \cdot 2}}$

Step 9.2.2

Multiply $3$ by $2$ .

$8 u^{6} v^{9} \frac{9 v^{2}}{u^{6}}$

Step 10

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

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Step 10.1

Factor $u^{6}$ out of $8 u^{6} v^{9}$ .

$u^{6} (8 v^{9}) \frac{9 v^{2}}{u^{6}}$

Step 10.2

Cancel the common factor.

$u^{6} (8 v^{9}) \frac{9 v^{2}}{u^{6}}$

Step 10.3

Rewrite the expression.

$8 v^{9} (9 v^{2})$

Step 11

Multiply $9$ by $8$ .

$72 v^{9} v^{2}$

Step 12

Multiply $v^{9}$ by $v^{2}$ by adding the exponents.

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Step 12.1

Move $v^{2}$ .

$72 (v^{2} v^{9})$

Step 12.2

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

$72 v^{2 + 9}$

Step 12.3

Add $2$ and $9$ .

$72 v^{11}$