Evaluate (-4mn)^3*(-2m^2)^3

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

Step 1

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

Apply the product rule to $- 4 m n$ .

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

Apply the product rule to $- 4 m$ .

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

Step 2

Raise $- 4$ to the power of $3$ .

$- 64 m^{3} n^{3} \cdot {(- 2 m^{2})}^{3}$

Step 3

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

$- 64 m^{3} n^{3} \cdot ({(- 2)}^{3} {(m^{2})}^{3})$

Step 4

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

$- 64 m^{3} n^{3} \cdot (- 8 {(m^{2})}^{3})$

Step 5

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

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

$- 64 m^{3} n^{3} \cdot (- 8 m^{2 \cdot 3})$

Multiply $2$ by $3$ .

$- 64 m^{3} n^{3} \cdot (- 8 m^{6})$

Step 6

Multiply $m^{3}$ by $m^{6}$ by adding the exponents.

Move $m^{6}$ .

$- 64 (m^{6} m^{3}) n^{3} \cdot - 8$

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

$- 64 m^{6 + 3} n^{3} \cdot - 8$

Add $6$ and $3$ .

$- 64 m^{9} n^{3} \cdot - 8$

Step 7

Multiply $- 8$ by $- 64$ .

$512 m^{9} n^{3}$