Pre-Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

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

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

Step 4

Cancel the common factor of $x - y$ .

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Step 4.1

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

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

Step 4.2

Cancel the common factor.

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

Step 4.3

Rewrite the expression.

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

Step 5

Multiply $\frac{x + y}{x y} \cdot \frac{1}{y}$ .

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Step 5.1

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

$\frac{x + y}{x y \cdot y}$

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

Cancel the common factor of $x$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Cancel the common factor of $y$ and $y^{2}$ .

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

Cancel the common factors.

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Step 8.3.1

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

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

Step 8.3.2

Cancel the common factor.

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

Step 8.3.3

Rewrite the expression.

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