Simplify (-3m^-2)^2(-2m^4n^2)^-1

${(- 3 m^{- 2})}^{2} {(- 2 m^{4} n^{2})}^{- 1}$

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

${(- 3 \frac{1}{m^{2}})}^{2} {(- 2 m^{4} n^{2})}^{- 1}$

Combine $- 3$ and $\frac{1}{m^{2}}$ .

${(\frac{- 3}{m^{2}})}^{2} {(- 2 m^{4} n^{2})}^{- 1}$

Move the negative in front of the fraction.

${(- \frac{3}{m^{2}})}^{2} {(- 2 m^{4} n^{2})}^{- 1}$

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

Apply the product rule to $- \frac{3}{m^{2}}$ .

${(- 1)}^{2} {(\frac{3}{m^{2}})}^{2} {(- 2 m^{4} n^{2})}^{- 1}$

Apply the product rule to $\frac{3}{m^{2}}$ .

${(- 1)}^{2} \frac{3^{2}}{{(m^{2})}^{2}} {(- 2 m^{4} n^{2})}^{- 1}$

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

$1 \frac{3^{2}}{{(m^{2})}^{2}} {(- 2 m^{4} n^{2})}^{- 1}$

Multiply $\frac{3^{2}}{{(m^{2})}^{2}}$ by $1$ .

$\frac{3^{2}}{{(m^{2})}^{2}} {(- 2 m^{4} n^{2})}^{- 1}$

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

$\frac{9}{{(m^{2})}^{2}} {(- 2 m^{4} n^{2})}^{- 1}$

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

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

$\frac{9}{m^{2 \cdot 2}} {(- 2 m^{4} n^{2})}^{- 1}$

Multiply $2$ by $2$ .

$\frac{9}{m^{4}} {(- 2 m^{4} n^{2})}^{- 1}$

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

$\frac{9}{m^{4}} \cdot \frac{1}{- 2 m^{4} n^{2}}$

Combine.

$\frac{9 \cdot 1}{m^{4} (- 2 m^{4} n^{2})}$

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

Move $m^{4}$ .

$\frac{9 \cdot 1}{m^{4} m^{4} (- 2 n^{2})}$

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

$\frac{9 \cdot 1}{m^{4 + 4} (- 2 n^{2})}$

Add $4$ and $4$ .

$\frac{9 \cdot 1}{m^{8} (- 2 n^{2})}$

Multiply $9$ by $1$ .

$\frac{9}{m^{8} (- 2 n^{2})}$

Simplify the denominator.

Rewrite.

$\frac{9}{m^{4} (- 2 m^{4} n^{2})}$

Remove unnecessary parentheses.

$\frac{9}{m^{8} \cdot - 2 n^{2}}$

Simplify the expression.

Move $- 2$ to the left of $m^{8}$ .

$\frac{9}{- 2 \cdot m^{8} n^{2}}$

Move the negative in front of the fraction.

$- \frac{9}{2 m^{8} n^{2}}$

Simplify (-3m^-2)^2(-2m^4n^2)^-1

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