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

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

Step 1

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

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

Step 2

Multiply $m$ by $m^{2}$ by adding the exponents.

Move $m^{2}$ .

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

Multiply $m^{2}$ by $m$ .

Raise $m$ to the power of $1$ .

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

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

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

Add $2$ and $1$ .

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

Step 3

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

Factor $n$ out of $4 m^{3} n$ .

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

Cancel the common factors.

Factor $n$ out of $n^{6}$ .

${(\frac{n (4 m^{3})}{n \cdot n^{5}})}^{- 2}$

Cancel the common factor.

${(\frac{n (4 m^{3})}{n \cdot n^{5}})}^{- 2}$

Rewrite the expression.

${(\frac{4 m^{3}}{n^{5}})}^{- 2}$

Step 4

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

${(\frac{n^{5}}{4 m^{3}})}^{2}$

Step 5

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

Apply the product rule to $\frac{n^{5}}{4 m^{3}}$ .

$\frac{{(n^{5})}^{2}}{{(4 m^{3})}^{2}}$

Apply the product rule to $4 m^{3}$ .

$\frac{{(n^{5})}^{2}}{4^{2} {(m^{3})}^{2}}$

Step 6

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

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

$\frac{n^{5 \cdot 2}}{4^{2} {(m^{3})}^{2}}$

Multiply $5$ by $2$ .

$\frac{n^{10}}{4^{2} {(m^{3})}^{2}}$

Step 7

Simplify the denominator.

Raise $4$ to the power of $2$ .

$\frac{n^{10}}{16 {(m^{3})}^{2}}$

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

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

$\frac{n^{10}}{16 m^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{n^{10}}{16 m^{6}}$

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

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