Simplify (a^4b)^-2(c^-3d)^2

${(a^{4} b)}^{- 2} {(c^{- 3} d)}^{2}$

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

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

Simplify the denominator.

Apply the product rule to $a^{4} b$ .

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

Multiply the exponents in ${(a^{4})}^{2}$ .

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

$\frac{1}{a^{4 \cdot 2} b^{2}} {(c^{- 3} d)}^{2}$

Multiply $4$ by $2$ .

$\frac{1}{a^{8} b^{2}} {(c^{- 3} d)}^{2}$

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

$\frac{1}{a^{8} b^{2}} {(\frac{1}{c^{3}} d)}^{2}$

Combine $\frac{1}{c^{3}}$ and $d$ .

$\frac{1}{a^{8} b^{2}} {(\frac{d}{c^{3}})}^{2}$

Apply the product rule to $\frac{d}{c^{3}}$ .

$\frac{1}{a^{8} b^{2}} \cdot \frac{d^{2}}{{(c^{3})}^{2}}$

Combine.

$\frac{1 d^{2}}{a^{8} b^{2} {(c^{3})}^{2}}$

Simplify the expression.

Multiply $d^{2}$ by $1$ .

$\frac{d^{2}}{a^{8} b^{2} {(c^{3})}^{2}}$

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

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

$\frac{d^{2}}{a^{8} b^{2} c^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$\frac{d^{2}}{a^{8} b^{2} c^{6}}$