Question

Simplify the complex rational expression by using the LCD:

$\frac{{\displaystyle \frac{1}{a}}+{\displaystyle \frac{1}{b}}}{{\displaystyle \frac{a}{b}}+{\displaystyle \frac{b}{a}}}$

Short answer

$\frac{a+b}{a^2+b^2}$

Step-by-step solution

Step 1. Given information.

The following complex rational expression is given:

$\frac{{\displaystyle \frac{1}{a}+\frac{1}{b}}}{{\displaystyle \frac{a}{b}+\frac{b}{a}}}$

Step 2. Simplification of the complex rational expression.

The LCD of all the fractions in the numerator and the denominator is $ab$.

Multiply the numerator and the denominator by that LCD:

$\frac{{\displaystyle ab.\left(\frac{1}{a}+\frac{1}{b}\right)}}{{\displaystyle ab.\left(\frac{a}{b}+\frac{b}{a}\right)}}$

Distribute:

$\frac{{\displaystyle ab.\frac{1}{a}+ab.\frac{1}{b}}}{{\displaystyle ab.\frac{a}{b}+ab.\frac{b}{a}}}$

Simplify:

$\frac{{\displaystyle b+a}}{{\displaystyle a^2+b^2}}$