Pre-Algebra Examples

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

Step 1

Cancel the common factor of $a^{- 7 b}$ and $a$ .

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Step 1.1

Factor $a$ out of $a^{- 7 b}$ .

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

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $a$ out of $a b^{5}$ .

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

Step 1.2.2

Cancel the common factor.

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

Step 1.2.3

Rewrite the expression.

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

Step 2

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

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

Step 3

Apply the product rule to $\frac{b^{5}}{a^{- 7 b - 1}}$ .

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

Step 4

Multiply the exponents in ${(b^{5})}^{2}$ .

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Step 4.1

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

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

Step 4.2

Multiply $5$ by $2$ .

$\frac{b^{10}}{{(a^{- 7 b - 1})}^{2}}$

Step 5

Multiply the exponents in ${(a^{- 7 b - 1})}^{2}$ .

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Step 5.1

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

$\frac{b^{10}}{a^{(- 7 b - 1) \cdot 2}}$

Step 5.2

Apply the distributive property.

$\frac{b^{10}}{a^{- 7 b \cdot 2 - 1 \cdot 2}}$

Step 5.3

Multiply $2$ by $- 7$ .

$\frac{b^{10}}{a^{- 14 b - 1 \cdot 2}}$

Step 5.4

Multiply $- 1$ by $2$ .

$\frac{b^{10}}{a^{- 14 b - 2}}$