Problem 63 - College Algebra

Question

Simplify each complex rational expression. $$\frac{\frac{1}{x}+\frac{1}{y}}{x+y}$$

Step-by-Step Solution

Answer

The simplified form of the given complex rational expression is \(\frac{1}{xy}\).

Step 1: Identify the Denominator

The denominator of the complex rational expression is \(x + y\), and this will be retained in the simplified form.

Step 2: Find the Common Denominator

In the numerator, the common denominator for the fractions \(\frac{1}{x}\) and \(\frac{1}{y}\) is \(x * y\) or \(xy\). Hence, you rewrite these fractions as \(\frac{y}{xy}\) and \(\frac{x}{xy}\), respectively.

Step 3: Simplify the Numerator

Add those fractions to simplify the numerator - which becomes \(\frac{x+y}{xy}\). So, the complex fraction can now be rewritten as \(\frac{(x+y)/xy}{x+y}\).

Step 4: Multiply the main Numerator and Denominator by the reciprocal

When you have a fraction within a fraction, you can multiply the main numerator and denominator by the reciprocal of the denominator. So, multiply the complex fraction by \( \frac{xy}{1}\), which simplifies to \(\frac{(x+y)}{(x+y)*xy} = \frac{1}{xy}\).