Simplify ((x^3y^-2)/(z^-5))^-4

${(\frac{x^{3} y^{- 2}}{z^{- 5}})}^{- 4}$

Step 1

Move $y^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{x^{3}}{z^{- 5} y^{2}})}^{- 4}$

Step 2

Move $z^{- 5}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{x^{3} z^{5}}{y^{2}})}^{- 4}$

Step 3

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

${(\frac{y^{2}}{x^{3} z^{5}})}^{4}$

Step 4

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

Apply the product rule to $\frac{y^{2}}{x^{3} z^{5}}$ .

$\frac{{(y^{2})}^{4}}{{(x^{3} z^{5})}^{4}}$

Apply the product rule to $x^{3} z^{5}$ .

$\frac{{(y^{2})}^{4}}{{(x^{3})}^{4} {(z^{5})}^{4}}$

Step 5

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

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

$\frac{y^{2 \cdot 4}}{{(x^{3})}^{4} {(z^{5})}^{4}}$

Multiply $2$ by $4$ .

$\frac{y^{8}}{{(x^{3})}^{4} {(z^{5})}^{4}}$

Step 6

Simplify the denominator.

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

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

$\frac{y^{8}}{x^{3 \cdot 4} {(z^{5})}^{4}}$

Multiply $3$ by $4$ .

$\frac{y^{8}}{x^{12} {(z^{5})}^{4}}$

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

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

$\frac{y^{8}}{x^{12} z^{5 \cdot 4}}$

Multiply $5$ by $4$ .

$\frac{y^{8}}{x^{12} z^{20}}$