Simplify ((-2mn^2)^-3)/(4m^-6n^4)

$\frac{{(- 2 m n^{2})}^{- 3}}{4 m^{- 6} n^{4}}$

Step 1

Simplify the expression.

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

$\frac{1}{4 m^{- 6} n^{4} {(- 2 m n^{2})}^{3}}$

Move $m^{- 6}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{m^{6}}{4 n^{4} {(- 2 m n^{2})}^{3}}$

Step 2

Simplify the denominator.

Apply the product rule to $- 2 m n^{2}$ .

$\frac{m^{6}}{4 n^{4} {(- 2 m)}^{3} {(n^{2})}^{3}}$

Apply the product rule to $- 2 m$ .

$\frac{m^{6}}{4 n^{4} ({(- 2)}^{3} m^{3}) {(n^{2})}^{3}}$

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

$\frac{m^{6}}{4 n^{4} (- 8 m^{3}) {(n^{2})}^{3}}$

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

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

$\frac{m^{6}}{4 n^{4} (- 8 m^{3}) n^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{m^{6}}{4 n^{4} (- 8 m^{3}) n^{6}}$

$\frac{m^{6}}{4 n^{4} \cdot - 8 m^{3} n^{6}}$

Combine exponents.

Multiply $- 8$ by $4$ .

$\frac{m^{6}}{- 32 n^{4} m^{3} n^{6}}$

Multiply $n^{4}$ by $n^{6}$ by adding the exponents.

Move $n^{6}$ .

$\frac{m^{6}}{- 32 (n^{6} n^{4}) m^{3}}$

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

$\frac{m^{6}}{- 32 n^{6 + 4} m^{3}}$

Add $6$ and $4$ .

$\frac{m^{6}}{- 32 n^{10} m^{3}}$

Step 3

Reduce the expression by cancelling the common factors.

Cancel the common factor of $m^{6}$ and $m^{3}$ .

Factor $m^{3}$ out of $m^{6}$ .

$\frac{m^{3} m^{3}}{- 32 n^{10} m^{3}}$

Cancel the common factors.

Factor $m^{3}$ out of $- 32 n^{10} m^{3}$ .

$\frac{m^{3} m^{3}}{m^{3} (- 32 n^{10})}$

Cancel the common factor.

$\frac{m^{3} m^{3}}{m^{3} (- 32 n^{10})}$

Rewrite the expression.

$\frac{m^{3}}{- 32 n^{10}}$

Move the negative in front of the fraction.

$- \frac{m^{3}}{32 n^{10}}$