Precalculus Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $- b$ .

$\frac{a^{5} {(- 1)}^{3} b^{3}}{{(a b)}^{2}} \cdot {(\frac{a}{2 b})}^{4}$

Step 1.2

Raise $- 1$ to the power of $3$ .

$\frac{a^{5} \cdot - 1 b^{3}}{{(a b)}^{2}} \cdot {(\frac{a}{2 b})}^{4}$

Step 1.3

Factor out negative.

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

Step 2

Apply the product rule to $a b$ .

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

Step 3

Cancel the common factor of $a^{5}$ and $a^{2}$ .

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Step 3.1

Factor $a^{2}$ out of $- a^{5} b^{3}$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $a^{2}$ out of $a^{2} b^{2}$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Cancel the common factor of $b^{3}$ and $b^{2}$ .

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Step 4.1

Factor $b^{2}$ out of $- a^{3} b^{3}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $- a^{3} b$ by $1$ .

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

Step 5

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

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Step 5.1

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

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

Step 5.2

Apply the product rule to $2 b$ .

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

Step 6

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

$- a^{3} b \cdot \frac{a^{4}}{16 b^{4}}$

Step 7

Cancel the common factor of $b$ .

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Step 7.1

Factor $b$ out of $- a^{3} b$ .

$b (- a^{3}) \cdot \frac{a^{4}}{16 b^{4}}$

Step 7.2

Factor $b$ out of $16 b^{4}$ .

$b (- a^{3}) \cdot \frac{a^{4}}{b (16 b^{3})}$

Step 7.3

Cancel the common factor.

$b (- a^{3}) \cdot \frac{a^{4}}{b (16 b^{3})}$

Step 7.4

Rewrite the expression.

$- a^{3} \cdot \frac{a^{4}}{16 b^{3}}$

Step 8

Combine $\frac{a^{4}}{16 b^{3}}$ and $a^{3}$ .

$- \frac{a^{4} a^{3}}{16 b^{3}}$

Step 9

Multiply $a^{4}$ by $a^{3}$ by adding the exponents.

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Step 9.1

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

$- \frac{a^{4 + 3}}{16 b^{3}}$

Step 9.2

Add $4$ and $3$ .

$- \frac{a^{7}}{16 b^{3}}$