Algebra Examples

$\frac{4 x y}{y^{2} - x^{2}} \div (\frac{1}{y^{2} - x^{2}} + \frac{1}{x^{2} + 2 x y + y^{2}})$

Step 1

Rewrite the division as a fraction.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{y^{2} - x^{2}} + \frac{1}{x^{2} + 2 x y + y^{2}}}$

Step 4

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

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{(y + x) (y - x)} + \frac{1}{x^{2} + 2 x y + y^{2}}}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Step 5.2

Rewrite the polynomial.

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{(y + x) (y - x)} + \frac{1}{x^{2} + 2 \cdot x \cdot y + y^{2}}}$

Step 5.3

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = y$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{(y + x) (y - x)} + \frac{1}{{(x + y)}^{2}}}$

Step 6

Simplify the denominator.

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Step 6.1

Reorder terms.

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{(x + y) (y - x)} + \frac{1}{{(x + y)}^{2}}}$

Step 6.2

To write $\frac{1}{(x + y) (y - x)}$ as a fraction with a common denominator, multiply by $\frac{x + y}{x + y}$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{(x + y) (y - x)} \cdot \frac{x + y}{x + y} + \frac{1}{{(x + y)}^{2}}}$

Step 6.3

To write $\frac{1}{{(x + y)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{y - x}{y - x}$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{1}{(x + y) (y - x)} \cdot \frac{x + y}{x + y} + \frac{1}{{(x + y)}^{2}} \cdot \frac{y - x}{y - x}}$

Step 6.4

Write each expression with a common denominator of $(y - x) {(x + y)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.4.1

Multiply $\frac{1}{(x + y) (y - x)}$ by $\frac{x + y}{x + y}$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y}{(x + y) (y - x) (x + y)} + \frac{1}{{(x + y)}^{2}} \cdot \frac{y - x}{y - x}}$

Step 6.4.2

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

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y}{{(x + y)}^{1} (x + y) (y - x)} + \frac{1}{{(x + y)}^{2}} \cdot \frac{y - x}{y - x}}$

Step 6.4.3

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

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y}{{(x + y)}^{1} {(x + y)}^{1} (y - x)} + \frac{1}{{(x + y)}^{2}} \cdot \frac{y - x}{y - x}}$

Step 6.4.4

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

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y}{{(x + y)}^{1 + 1} (y - x)} + \frac{1}{{(x + y)}^{2}} \cdot \frac{y - x}{y - x}}$

Step 6.4.5

Add $1$ and $1$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y}{{(x + y)}^{2} (y - x)} + \frac{1}{{(x + y)}^{2}} \cdot \frac{y - x}{y - x}}$

Step 6.4.6

Multiply $\frac{1}{{(x + y)}^{2}}$ by $\frac{y - x}{y - x}$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y}{{(x + y)}^{2} (y - x)} + \frac{y - x}{{(x + y)}^{2} (y - x)}}$

Step 6.5

Combine the numerators over the common denominator.

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{x + y + y - x}{{(x + y)}^{2} (y - x)}}$

Step 6.6

Rewrite $\frac{x + y + y - x}{{(x + y)}^{2} (y - x)}$ in a factored form.

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Step 6.6.1

Subtract $x$ from $x$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{y + y + 0}{{(x + y)}^{2} (y - x)}}$

Step 6.6.2

Add $y + y$ and $0$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{y + y}{{(x + y)}^{2} (y - x)}}$

Step 6.6.3

Add $y$ and $y$ .

$\frac{4 x y}{(y + x) (y - x)} \cdot \frac{1}{\frac{2 y}{{(x + y)}^{2} (y - x)}}$

Step 7

Combine fractions.

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Step 7.1

Combine.

$\frac{4 x y \cdot 1}{(y + x) (y - x) \frac{2 y}{{(x + y)}^{2} (y - x)}}$

Step 7.2

Multiply $4$ by $1$ .

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

Step 8

Factor $\frac{2 y}{{(x + y)}^{2} (y - x)}$ out of $\frac{4 x y}{(y + x) (y - x) \frac{2 y}{{(x + y)}^{2} (y - x)}}$ .

$\frac{{(x + y)}^{2} (y - x)}{2 y} \cdot \frac{4 x y}{(y + x) (y - x)}$

Step 9

Combine.

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

Step 10

Cancel the common factor of $y - x$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

Cancel the common factor of $4$ and $2$ .

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Step 11.1

Factor $2$ out of ${(x + y)}^{2} (4 x y)$ .

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

Step 11.2

Cancel the common factors.

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Step 11.2.1

Factor $2$ out of $2 y (y + x)$ .

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

Step 11.2.2

Cancel the common factor.

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

Step 11.2.3

Rewrite the expression.

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

Step 12

Cancel the common factor of $y$ .

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Step 12.1

Cancel the common factor.

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

Step 12.2

Rewrite the expression.

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

Step 13

Cancel the common factor of ${(x + y)}^{2}$ and $y + x$ .

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Step 13.1

Reorder terms.

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

Step 13.2

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

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

Step 13.3

Cancel the common factors.

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Step 13.3.1

Multiply by $1$ .

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

Step 13.3.2

Cancel the common factor.

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

Step 13.3.3

Rewrite the expression.

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

Step 13.3.4

Divide $(x + y) (2 x)$ by $1$ .

$(x + y) (2 x)$

Step 14

Apply the distributive property.

$x (2 x) + y (2 x)$

Step 15

Reorder.

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Step 15.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + y (2 x)$

Step 15.2

Rewrite using the commutative property of multiplication.

$2 x \cdot x + 2 y x$

Step 16

Multiply $x$ by $x$ by adding the exponents.

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Step 16.1

Move $x$ .

$2 (x \cdot x) + 2 y x$

Step 16.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 y x$