Simplify ((3xy^-2)/(x^3))^2

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

Step 1

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

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

Step 2

Cancel the common factor of $x$ and $x^{3}$ .

Factor $x$ out of $3 x$ .

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

Cancel the common factors.

Factor $x$ out of $x^{3} y^{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

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

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

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

Step 4

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

$\frac{9}{{(x^{2})}^{2} {(y^{2})}^{2}}$

Step 5

Simplify the denominator.

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

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

$\frac{9}{x^{2 \cdot 2} {(y^{2})}^{2}}$

Multiply $2$ by $2$ .

$\frac{9}{x^{4} {(y^{2})}^{2}}$

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

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

$\frac{9}{x^{4} y^{2 \cdot 2}}$

Multiply $2$ by $2$ .

$\frac{9}{x^{4} y^{4}}$