Simplify (4v^3w)(-2w^3)^2

$(4 v^{3} w) {(- 2 w^{3})}^{2}$

Apply the product rule to $- 2 w^{3}$ .

$4 v^{3} w ({(- 2)}^{2} {(w^{3})}^{2})$

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

$4 v^{3} w (4 {(w^{3})}^{2})$

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

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

$4 v^{3} w (4 w^{3 \cdot 2})$

Multiply $3$ by $2$ .

$4 v^{3} w (4 w^{6})$

Multiply $w$ by $w^{6}$ by adding the exponents.

Move $w^{6}$ .

$4 v^{3} (w^{6} w) \cdot 4$

Multiply $w^{6}$ by $w$ .

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

$4 v^{3} (w^{6} w^{1}) \cdot 4$

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

$4 v^{3} w^{6 + 1} \cdot 4$

Add $6$ and $1$ .

$4 v^{3} w^{7} \cdot 4$

Multiply $4$ by $4$ .

$16 v^{3} w^{7}$