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

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

Step 1

Simplify.

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

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

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

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

Step 2

Simplify the numerator.

Apply the product rule to $2 m n$ .

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

Multiply $n^{4}$ by $n^{2}$ by adding the exponents.

Move $n^{2}$ .

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

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

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

Add $2$ and $4$ .

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

Apply the product rule to $2 m$ .

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

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

$\frac{16 m^{4} n^{6} m^{3}}{6}$

Multiply $m^{4}$ by $m^{3}$ by adding the exponents.

Move $m^{3}$ .

$\frac{16 (m^{3} m^{4}) n^{6}}{6}$

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

$\frac{16 m^{3 + 4} n^{6}}{6}$

Add $3$ and $4$ .

$\frac{16 m^{7} n^{6}}{6}$

Step 3

Cancel the common factor of $16$ and $6$ .

Factor $2$ out of $16 m^{7} n^{6}$ .

$\frac{2 (8 m^{7} n^{6})}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 (8 m^{7} n^{6})}{2 (3)}$

Cancel the common factor.

$\frac{2 (8 m^{7} n^{6})}{2 \cdot 3}$

Rewrite the expression.

$\frac{8 m^{7} n^{6}}{3}$