Simplify (-4a^2)(2a^-3)^-4

$(- 4 a^{2}) {(2 a^{- 3})}^{- 4}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 4 a^{2} {(2 \frac{1}{a^{3}})}^{- 4}$

Combine $2$ and $\frac{1}{a^{3}}$ .

$- 4 a^{2} {(\frac{2}{a^{3}})}^{- 4}$

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

$- 4 a^{2} {(\frac{a^{3}}{2})}^{4}$

Simplify the expression.

Apply the product rule to $\frac{a^{3}}{2}$ .

$- 4 a^{2} \frac{{(a^{3})}^{4}}{2^{4}}$

Multiply the exponents in ${(a^{3})}^{4}$ .

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

$- 4 a^{2} \frac{a^{3 \cdot 4}}{2^{4}}$

Multiply $3$ by $4$ .

$- 4 a^{2} \frac{a^{12}}{2^{4}}$

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

$- 4 a^{2} \frac{a^{12}}{16}$

Cancel the common factor of $4$ .

Factor $4$ out of $- 4 a^{2}$ .

$4 (- a^{2}) \frac{a^{12}}{16}$

Factor $4$ out of $16$ .

$4 (- a^{2}) \frac{a^{12}}{4 (4)}$

Cancel the common factor.

$4 (- a^{2}) \frac{a^{12}}{4 \cdot 4}$

Rewrite the expression.

$- a^{2} \frac{a^{12}}{4}$

Combine $\frac{a^{12}}{4}$ and $a^{2}$ .

$- \frac{a^{12} a^{2}}{4}$

Multiply $a^{12}$ by $a^{2}$ by adding the exponents.

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

$- \frac{a^{12 + 2}}{4}$

Add $12$ and $2$ .

$- \frac{a^{14}}{4}$