1Cademy - Try It 7.58: Simplifying $$\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x}+\frac{1}{y}}$$ Using the LCD

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Simplifying a Complex Rational Expression by Using the LCD

Example

Try It 7.58: Simplifying $\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x}+\frac{1}{y}}$ Using the LCD

To simplify the complex rational expression $\frac{\frac{1}{x^2}-\frac{1}{y^2}}{\frac{1}{x}+\frac{1}{y}}$ using the least common denominator (LCD) method:

Step 1. Find the LCD of all inner fractions. The denominators are $x^2$ , $y^2$ , $x$ , and $y$ . The LCD is $x^2y^2$ .

Step 2. Multiply the numerator and denominator by the LCD, $x^2y^2$ .

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

Step 3. Simplify by canceling common factors in each term:

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

Step 4. Factor both the numerator and the denominator completely. The numerator is a difference of squares: $(y - x)(y + x)$ . The denominator has a greatest common factor of $xy$ : $xy(y + x)$ .