Algebra Examples

$x y - \frac{1}{x y}$

Step 1

To write $x y$ as a fraction with a common denominator, multiply by $\frac{x y}{x y}$ .

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

Step 2

Simplify terms.

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Step 2.1

Combine $x y$ and $\frac{x y}{x y}$ .

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $1$ and $1$ .

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

Step 3.9

Rewrite $x^{2} y^{2}$ as ${(x y)}^{2}$ .

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

Step 3.10

Rewrite $1$ as $1^{2}$ .

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

Step 3.11

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x y$ and $b = 1$ .

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