Simplify ((x^-2y^-3)^-2)/((6x^-2y^-1)^-1)

$\frac{{(x^{- 2} y^{- 3})}^{- 2}}{{(6 x^{- 2} y^{- 1})}^{- 1}}$

Step 1

Simplify the expression.

Move ${(x^{- 2} y^{- 3})}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(6 x^{- 2} y^{- 1})}^{- 1} {(x^{- 2} y^{- 3})}^{2}}$

Move ${(6 x^{- 2} y^{- 1})}^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$\frac{6 x^{- 2} y^{- 1}}{{(x^{- 2} y^{- 3})}^{2}}$

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

$\frac{6 y^{- 1}}{{(x^{- 2} y^{- 3})}^{2} x^{2}}$

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

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

Step 2

Simplify the denominator.

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

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

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

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

$\frac{6}{x^{- 2 \cdot 2} {(y^{- 3})}^{2} x^{2} y}$

Multiply $- 2$ by $2$ .

$\frac{6}{x^{- 4} {(y^{- 3})}^{2} x^{2} y}$

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

$\frac{6}{\frac{1}{x^{4}} {(y^{- 3})}^{2} x^{2} y}$

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

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

$\frac{6}{\frac{1}{x^{4}} y^{- 3 \cdot 2} x^{2} y}$

Multiply $- 3$ by $2$ .

$\frac{6}{\frac{1}{x^{4}} y^{- 6} x^{2} y}$

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

$\frac{6}{\frac{1}{x^{4}} \cdot \frac{1}{y^{6}} x^{2} y}$

Combine exponents.

Multiply $\frac{1}{x^{4}}$ by $\frac{1}{y^{6}}$ .

$\frac{6}{\frac{1}{x^{4} y^{6}} x^{2} y}$

Combine $\frac{1}{x^{4} y^{6}}$ and $x^{2}$ .

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

Combine $\frac{x^{2}}{x^{4} y^{6}}$ and $y$ .

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

Reduce the expression $\frac{x^{2} y}{x^{4} y^{6}}$ by cancelling the common factors.

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

$\frac{6}{\frac{x^{2} (y)}{x^{4} y^{6}}}$

Factor $x^{2}$ out of $x^{4} y^{6}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{6}{\frac{y}{x^{2} y^{6}}}$

Cancel the common factor of $y$ and $y^{6}$ .

Raise $y$ to the power of $1$ .

$\frac{6}{\frac{y^{1}}{x^{2} y^{6}}}$

Factor $y$ out of $y^{1}$ .

$\frac{6}{\frac{y \cdot 1}{x^{2} y^{6}}}$

Cancel the common factors.

Factor $y$ out of $x^{2} y^{6}$ .

$\frac{6}{\frac{y \cdot 1}{y (x^{2} y^{5})}}$

Cancel the common factor.

$\frac{6}{\frac{y \cdot 1}{y (x^{2} y^{5})}}$

Rewrite the expression.

$\frac{6}{\frac{1}{x^{2} y^{5}}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$6 x^{2} y^{5}$

Simplify ((x^-2y^-3)^-2)/((6x^-2y^-1)^-1)

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