Simplify (-2a^-2b^-1)^2

${(- 2 a^{- 2} b^{- 1})}^{2}$

Step 1

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

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

Step 2

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

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

Step 3

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

Multiply $\frac{1}{b}$ by $\frac{2}{a^{2}}$ .

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

Step 6

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

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

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

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

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

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

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

Step 7

Raise $- 1$ to the power of $2$ .

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

Step 8

Multiply $\frac{2^{2}}{b^{2} {(a^{2})}^{2}}$ by $1$ .

$\frac{2^{2}}{b^{2} {(a^{2})}^{2}}$

Step 9

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

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

Step 10

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

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

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

Multiply $2$ by $2$ .

$\frac{4}{b^{2} a^{4}}$