Simplify ((p^-3q^-2)/(q^-3r^5))^4

${(\frac{p^{- 3} q^{- 2}}{q^{- 3} r^{5}})}^{4}$

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

${(\frac{q^{- 2}}{q^{- 3} r^{5} p^{3}})}^{4}$

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

${(\frac{q^{- 2} q^{3}}{r^{5} p^{3}})}^{4}$

Simplify the numerator.

Multiply $q^{- 2}$ by $q^{3}$ by adding the exponents.

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

${(\frac{q^{- 2 + 3}}{r^{5} p^{3}})}^{4}$

Add $- 2$ and $3$ .

${(\frac{q^{1}}{r^{5} p^{3}})}^{4}$

Simplify $q^{1}$ .

${(\frac{q}{r^{5} p^{3}})}^{4}$

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

Apply the product rule to $\frac{q}{r^{5} p^{3}}$ .

$\frac{q^{4}}{{(r^{5} p^{3})}^{4}}$

Apply the product rule to $r^{5} p^{3}$ .

$\frac{q^{4}}{{(r^{5})}^{4} {(p^{3})}^{4}}$

Simplify the denominator.

Multiply the exponents in ${(r^{5})}^{4}$ .

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

$\frac{q^{4}}{r^{5 \cdot 4} {(p^{3})}^{4}}$

Multiply $5$ by $4$ .

$\frac{q^{4}}{r^{20} {(p^{3})}^{4}}$

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

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

$\frac{q^{4}}{r^{20} p^{3 \cdot 4}}$

Multiply $3$ by $4$ .

$\frac{q^{4}}{r^{20} p^{12}}$