Precalculus Examples

${(\frac{2 a^{- 1} b}{a^{3} b^{- 4}})}^{- 3}$

Step 1

Move $a^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{2 b}{a^{3} b^{- 4} a})}^{- 3}$

Step 2

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 2.1

Move $a$ .

${(\frac{2 b}{a \cdot a^{3} b^{- 4}})}^{- 3}$

Step 2.2

Multiply $a$ by $a^{3}$ .

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Step 2.2.1

Raise $a$ to the power of $1$ .

${(\frac{2 b}{a^{1} a^{3} b^{- 4}})}^{- 3}$

Step 2.2.2

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

${(\frac{2 b}{a^{1 + 3} b^{- 4}})}^{- 3}$

Step 2.3

Add $1$ and $3$ .

${(\frac{2 b}{a^{4} b^{- 4}})}^{- 3}$

Step 3

Move $b^{- 4}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{2 b \cdot b^{4}}{a^{4}})}^{- 3}$

Step 4

Multiply $b$ by $b^{4}$ by adding the exponents.

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Step 4.1

Move $b^{4}$ .

${(\frac{2 (b^{4} b)}{a^{4}})}^{- 3}$

Step 4.2

Multiply $b^{4}$ by $b$ .

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Step 4.2.1

Raise $b$ to the power of $1$ .

${(\frac{2 (b^{4} b^{1})}{a^{4}})}^{- 3}$

Step 4.2.2

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

${(\frac{2 b^{4 + 1}}{a^{4}})}^{- 3}$

Step 4.3

Add $4$ and $1$ .

${(\frac{2 b^{5}}{a^{4}})}^{- 3}$

Step 5

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{a^{4}}{2 b^{5}})}^{3}$

Step 6

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

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Step 6.1

Apply the product rule to $\frac{a^{4}}{2 b^{5}}$ .

$\frac{{(a^{4})}^{3}}{{(2 b^{5})}^{3}}$

Step 6.2

Apply the product rule to $2 b^{5}$ .

$\frac{{(a^{4})}^{3}}{2^{3} {(b^{5})}^{3}}$

Step 7

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

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Step 7.1

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

$\frac{a^{4 \cdot 3}}{2^{3} {(b^{5})}^{3}}$

Step 7.2

Multiply $4$ by $3$ .

$\frac{a^{12}}{2^{3} {(b^{5})}^{3}}$

Step 8

Simplify the denominator.

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Step 8.1

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

$\frac{a^{12}}{8 {(b^{5})}^{3}}$

Step 8.2

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

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Step 8.2.1

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

$\frac{a^{12}}{8 b^{5 \cdot 3}}$

Step 8.2.2

Multiply $5$ by $3$ .

$\frac{a^{12}}{8 b^{15}}$