Algebra Examples

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

Step 1

Move ${(2 a^{- 2} b)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Move $b^{3}$ .

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

Step 2.2.2

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

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

Step 2.2.3

Add $3$ and $4$ .

$\frac{1}{5 a^{2} b^{7} {(2 a^{- 2})}^{3}}$

Step 2.3

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

$\frac{1}{5 a^{2} b^{7} {(2 \frac{1}{a^{2}})}^{3}}$

Step 2.4

Combine $2$ and $\frac{1}{a^{2}}$ .

$\frac{1}{5 a^{2} b^{7} {(\frac{2}{a^{2}})}^{3}}$

Step 2.5

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

$\frac{1}{5 a^{2} b^{7} \frac{2^{3}}{{(a^{2})}^{3}}}$

Step 2.6

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

$\frac{1}{5 a^{2} b^{7} \frac{8}{{(a^{2})}^{3}}}$

Step 2.7

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

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Step 2.7.1

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

$\frac{1}{5 a^{2} b^{7} \frac{8}{a^{2 \cdot 3}}}$

Step 2.7.2

Multiply $2$ by $3$ .

$\frac{1}{5 a^{2} b^{7} \frac{8}{a^{6}}}$

Step 2.8

Combine exponents.

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Step 2.8.1

Combine $5$ and $\frac{8}{a^{6}}$ .

$\frac{1}{a^{2} b^{7} \frac{5 \cdot 8}{a^{6}}}$

Step 2.8.2

Multiply $5$ by $8$ .

$\frac{1}{a^{2} b^{7} \frac{40}{a^{6}}}$

Step 2.8.3

Combine $a^{2}$ and $\frac{40}{a^{6}}$ .

$\frac{1}{b^{7} \frac{a^{2} \cdot 40}{a^{6}}}$

Step 2.8.4

Combine $b^{7}$ and $\frac{a^{2} \cdot 40}{a^{6}}$ .

$\frac{1}{\frac{b^{7} (a^{2} \cdot 40)}{a^{6}}}$

Step 2.9

Remove unnecessary parentheses.

$\frac{1}{\frac{b^{7} a^{2} \cdot 40}{a^{6}}}$

Step 2.10

Reduce the expression $\frac{b^{7} a^{2} \cdot 40}{a^{6}}$ by cancelling the common factors.

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Step 2.10.1

Factor $a^{2}$ out of $b^{7} a^{2} \cdot 40$ .

$\frac{1}{\frac{a^{2} (b^{7} \cdot 40)}{a^{6}}}$

Step 2.10.2

Factor $a^{2}$ out of $a^{6}$ .

$\frac{1}{\frac{a^{2} (b^{7} \cdot 40)}{a^{2} a^{4}}}$

Step 2.10.3

Cancel the common factor.

$\frac{1}{\frac{a^{2} (b^{7} \cdot 40)}{a^{2} a^{4}}}$

Step 2.10.4

Rewrite the expression.

$\frac{1}{\frac{b^{7} \cdot 40}{a^{4}}}$

Step 2.11

Move $40$ to the left of $b^{7}$ .

$\frac{1}{\frac{40 b^{7}}{a^{4}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{a^{4}}{40 b^{7}}$

Step 4

Multiply $\frac{a^{4}}{40 b^{7}}$ by $1$ .

$\frac{a^{4}}{40 b^{7}}$