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

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

Step 1

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

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

Step 2

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

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

Step 3

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

Apply the product rule to $\frac{z^{4} y^{2}}{2 x^{2}}$ .

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

Apply the product rule to $z^{4} y^{2}$ .

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

Apply the product rule to $2 x^{2}$ .

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

Step 4

Simplify the numerator.

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

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

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

Multiply $4$ by $3$ .

$\frac{z^{12} {(y^{2})}^{3}}{2^{3} {(x^{2})}^{3}}$

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

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

$\frac{z^{12} y^{2 \cdot 3}}{2^{3} {(x^{2})}^{3}}$

Multiply $2$ by $3$ .

$\frac{z^{12} y^{6}}{2^{3} {(x^{2})}^{3}}$

Step 5

Simplify the denominator.

Raise $2$ to the power of $3$ .

$\frac{z^{12} y^{6}}{8 {(x^{2})}^{3}}$

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

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

$\frac{z^{12} y^{6}}{8 x^{2 \cdot 3}}$

Multiply $2$ by $3$ .

$\frac{z^{12} y^{6}}{8 x^{6}}$