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

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

Step 1

Cancel the common factor of $x^{3}$ and $x^{- 2}$ .

Factor $x^{- 2}$ out of $x^{3} y^{- 2} {(x + y)}^{- 1}$ .

$\frac{\frac{{(x + y)}^{- 2}}{y^{- 3}}}{\frac{x^{- 2} (x^{5} y^{- 2} {(x + y)}^{- 1})}{x^{- 2}}}$

Cancel the common factors.

Multiply by $1$ .

$\frac{\frac{{(x + y)}^{- 2}}{y^{- 3}}}{\frac{x^{- 2} (x^{5} y^{- 2} {(x + y)}^{- 1})}{x^{- 2} \cdot 1}}$

Cancel the common factor.

$\frac{\frac{{(x + y)}^{- 2}}{y^{- 3}}}{\frac{x^{- 2} (x^{5} y^{- 2} {(x + y)}^{- 1})}{x^{- 2} \cdot 1}}$

Rewrite the expression.

$\frac{\frac{{(x + y)}^{- 2}}{y^{- 3}}}{\frac{x^{5} y^{- 2} {(x + y)}^{- 1}}{1}}$

Divide $x^{5} y^{- 2} {(x + y)}^{- 1}$ by $1$ .

$\frac{\frac{{(x + y)}^{- 2}}{y^{- 3}}}{x^{5} y^{- 2} {(x + y)}^{- 1}}$

Step 2

Simplify the expression.

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

$\frac{\frac{{(x + y)}^{- 2}}{y^{- 3}} y^{2}}{x^{5} {(x + y)}^{- 1}}$

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

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

Step 3

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

$\frac{\frac{{(x + y)}^{- 2}}{\frac{1}{y^{3}}} y^{2} (x + y)}{x^{5}}$

Step 4

Combine $\frac{{(x + y)}^{- 2}}{\frac{1}{y^{3}}}$ and $y^{2}$ .

$\frac{\frac{{(x + y)}^{- 2} y^{2}}{\frac{1}{y^{3}}} (x + y)}{x^{5}}$

Step 5

Simplify the numerator.

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

$\frac{\frac{y^{2}}{\frac{1}{y^{3}} {(x + y)}^{2}} (x + y)}{x^{5}}$

Combine $\frac{1}{y^{3}}$ and ${(x + y)}^{2}$ .

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

Multiply the numerator by the reciprocal of the denominator.

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

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

Combine $y^{2}$ and $\frac{y^{3}}{{(x + y)}^{2}}$ .

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

Multiply $y^{2}$ by $y^{3}$ by adding the exponents.

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

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

Add $2$ and $3$ .

$\frac{\frac{y^{5}}{{(x + y)}^{2}} (x + y)}{x^{5}}$

Step 6

Simplify terms.

Factor $\frac{y^{5}}{{(x + y)}^{2}}$ out of $\frac{\frac{y^{5}}{{(x + y)}^{2}} (x + y)}{x^{5}}$ .

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

Cancel the common factor of $x + y$ .

Factor $x + y$ out of ${(x + y)}^{2}$ .

$\frac{y^{5}}{(x + y) (x + y)} \cdot \frac{x + y}{x^{5}}$

Cancel the common factor.

$\frac{y^{5}}{(x + y) (x + y)} \cdot \frac{x + y}{x^{5}}$

Rewrite the expression.

$\frac{y^{5}}{x + y} \cdot \frac{1}{x^{5}}$

Multiply $\frac{y^{5}}{x + y}$ by $\frac{1}{x^{5}}$ .

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

Reorder factors in $\frac{y^{5}}{(x + y) x^{5}}$ .

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

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

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