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

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

Step 1

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

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

Step 2

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

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

Step 3

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

Apply the product rule to $\frac{4 y^{3}}{3 z^{5} x^{5}}$ .

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

Apply the product rule to $4 y^{3}$ .

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

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

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

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

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

Step 4

Simplify the numerator.

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

$\frac{16 {(y^{3})}^{2}}{3^{2} {(z^{5})}^{2} {(x^{5})}^{2}}$

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

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

$\frac{16 y^{3 \cdot 2}}{3^{2} {(z^{5})}^{2} {(x^{5})}^{2}}$

Multiply $3$ by $2$ .

$\frac{16 y^{6}}{3^{2} {(z^{5})}^{2} {(x^{5})}^{2}}$

Step 5

Simplify the denominator.

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

$\frac{16 y^{6}}{9 {(z^{5})}^{2} {(x^{5})}^{2}}$

Multiply the exponents in ${(z^{5})}^{2}$ .

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

$\frac{16 y^{6}}{9 z^{5 \cdot 2} {(x^{5})}^{2}}$

Multiply $5$ by $2$ .

$\frac{16 y^{6}}{9 z^{10} {(x^{5})}^{2}}$

Multiply the exponents in ${(x^{5})}^{2}$ .

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

$\frac{16 y^{6}}{9 z^{10} x^{5 \cdot 2}}$

Multiply $5$ by $2$ .

$\frac{16 y^{6}}{9 z^{10} x^{10}}$