Question

Simplify the complex rational expression by using the LCD:

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

Short answer

The simplified answer is$\frac{y-x}{xy}$

Step-by-step solution

Step 1. Given information.

The following complex rational expression is given:

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

Step 2. Simplification of the complex rational expression.

The LCD of all the fractions in the numerator and the denominator is $x^2{y}^2$.

Multiply both the numerator and the denominator by the LCD:

$\frac{{\displaystyle x^2{y}^2.\left(\frac{1}{x^2}-\frac{1}{y^2}\right)}}{{\displaystyle x^2{y}^2.\left(\frac{1}{x}+\frac{1}{y}\right)}}$

Distribute:

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

Simplify:

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

Factor the numerator and the denominator, c cancel the common terms and simplify:

$$\begin{gathered} \frac{{\displaystyle \left(y+x\right)\left(y-x\right)}}{{\displaystyle xy\left(y+x\right)}} \\ \frac{y-x}{xy} \end{gathered}$$