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

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

Step 1

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

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

Step 2

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

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

Step 3

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

Multiply $\frac{1}{x^{2}}$ by $\frac{y^{2}}{y^{2}}$ .

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

Multiply $\frac{1}{y^{2}}$ by $\frac{x^{2}}{x^{2}}$ .

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

Reorder the factors of $y^{2} x^{2}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

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 $b = x$ .

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