Evaluate ((a^-3)/(b^2))^4

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

Step 1

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

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

Step 2

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

Apply the product rule to $\frac{1}{b^{2} a^{3}}$ .

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

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

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

Step 3

One to any power is one.

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

Step 4

Simplify the denominator.

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

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

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

Multiply $2$ by $4$ .

$\frac{1}{b^{8} {(a^{3})}^{4}}$

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

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

$\frac{1}{b^{8} a^{3 \cdot 4}}$

Multiply $3$ by $4$ .

$\frac{1}{b^{8} a^{12}}$