Simplify ((xy)^-3)/((x^-5y)^3)

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

Step 1

Move ${(x y)}^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(x^{- 5} y)}^{3} {(x y)}^{3}}$

Step 2

Simplify the denominator.

Apply the product rule to $x^{- 5} y$ .

$\frac{1}{{(x^{- 5})}^{3} y^{3} {(x y)}^{3}}$

Apply the product rule to $x y$ .

$\frac{1}{{(x^{- 5})}^{3} y^{3} x^{3} y^{3}}$

Multiply $y^{3}$ by $y^{3}$ by adding the exponents.

Move $y^{3}$ .

$\frac{1}{{(x^{- 5})}^{3} (y^{3} y^{3}) x^{3}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{{(x^{- 5})}^{3} y^{3 + 3} x^{3}}$

Add $3$ and $3$ .

$\frac{1}{{(x^{- 5})}^{3} y^{6} x^{3}}$

Multiply the exponents in ${(x^{- 5})}^{3}$ .

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

$\frac{1}{x^{- 5 \cdot 3} y^{6} x^{3}}$

Multiply $- 5$ by $3$ .

$\frac{1}{x^{- 15} y^{6} x^{3}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{\frac{1}{x^{15}} y^{6} x^{3}}$

Combine exponents.

Combine $\frac{1}{x^{15}}$ and $y^{6}$ .

$\frac{1}{\frac{y^{6}}{x^{15}} x^{3}}$

Combine $\frac{y^{6}}{x^{15}}$ and $x^{3}$ .

$\frac{1}{\frac{y^{6} x^{3}}{x^{15}}}$

Reduce the expression $\frac{y^{6} x^{3}}{x^{15}}$ by cancelling the common factors.

Factor $x^{3}$ out of $y^{6} x^{3}$ .

$\frac{1}{\frac{x^{3} y^{6}}{x^{15}}}$

Factor $x^{3}$ out of $x^{15}$ .

$\frac{1}{\frac{x^{3} y^{6}}{x^{3} x^{12}}}$

Cancel the common factor.

$\frac{1}{\frac{x^{3} y^{6}}{x^{3} x^{12}}}$

Rewrite the expression.

$\frac{1}{\frac{y^{6}}{x^{12}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{x^{12}}{y^{6}}$

Step 4

Multiply $\frac{x^{12}}{y^{6}}$ by $1$ .

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