Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

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

Step 3

Factor $3$ out of $3 x + 3 y$ .

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Step 3.1

Factor $3$ out of $3 x$ .

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

Step 3.2

Factor $3$ out of $3 y$ .

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

Step 3.3

Factor $3$ out of $3 (x) + 3 (y)$ .

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

Step 4

Cancel the common factor of $x + y$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

Multiply ${(x - y)}^{2}$ by $x - y$ by adding the exponents.

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Step 6.1

Multiply ${(x - y)}^{2}$ by $x - y$ .

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

Add $2$ and $1$ .

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

Step 7

Use the Binomial Theorem.

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

Step 8

Simplify each term.

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Step 8.1

Rewrite using the commutative property of multiplication.

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

Step 8.2

Multiply $3$ by $- 1$ .

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

Step 8.3

Apply the product rule to $- y$ .

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

Step 8.4

Rewrite using the commutative property of multiplication.

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

Step 8.5

Raise $- 1$ to the power of $2$ .

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

Step 8.6

Multiply $3$ by $1$ .

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

Step 8.7

Apply the product rule to $- y$ .

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

Step 8.8

Raise $- 1$ to the power of $3$ .

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

Step 9

Split the fraction $\frac{x^{3} - 3 x^{2} y + 3 x y^{2} - y^{3}}{3 (x + y)}$ into two fractions.

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

Step 10

Split the fraction $\frac{x^{3} - 3 x^{2} y + 3 x y^{2}}{3 (x + y)}$ into two fractions.

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

Step 11

Split the fraction $\frac{x^{3} - 3 x^{2} y}{3 (x + y)}$ into two fractions.

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

Step 12

Cancel the common factor of $- 3$ and $3$ .

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Step 12.1

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

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

Step 12.2

Cancel the common factors.

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Step 12.2.1

Cancel the common factor.

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

Step 12.2.2

Rewrite the expression.

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

Step 13

Move the negative in front of the fraction.

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

Step 14

Cancel the common factor of $3$ .

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Step 14.1

Cancel the common factor.

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

Step 14.2

Rewrite the expression.

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

Step 15

Move the negative in front of the fraction.

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