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Chapter 0: Problem 63

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

Short Answer

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

Step by step solution

01

Identify the Denominator

The denominator of the complex rational expression is \(x + y\), and this will be retained in the simplified form.
02

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.
03

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 \(\frac{(x+y)/xy}{x+y}\).
04

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 \(\frac{(x+y)}{(x+y)*xy} = \frac{1}{xy}\).

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