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

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

Step 1

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

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

Step 2

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

Apply the product rule to $\frac{4 z}{x^{3} y^{2}}$ .

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

Apply the product rule to $4 z$ .

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

Apply the product rule to $x^{3} y^{2}$ .

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

Step 3

Raise $4$ to the power of $2$ .

$\frac{16 z^{2}}{{(x^{3})}^{2} {(y^{2})}^{2}}$

Step 4

Simplify the denominator.

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

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

$\frac{16 z^{2}}{x^{3 \cdot 2} {(y^{2})}^{2}}$

Multiply $3$ by $2$ .

$\frac{16 z^{2}}{x^{6} {(y^{2})}^{2}}$

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

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

$\frac{16 z^{2}}{x^{6} y^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{16 z^{2}}{x^{6} y^{4}}$