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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 complex rational expression \(\frac{\frac{1}{x}+\frac{1}{y}}{x+y}\) is \(\frac{1}{xy}\).

Step by step solution

01

Find the LCD for the fractions in the numerator.

Identify the fractions present in the numerator of the complex fraction, which are \( \frac{1}{x} \) and \( \frac{1}{y} \). The least common denominator (LCD) of these fractions is the product of the denominators, that is, \( xy \).
02

Rewrite the numerator using the LCD.

Rewrite each fraction in the numerator using the least common denominator (LCD). Multiply each term by the LCD / LCD and simplify to get \(\frac{y}{xy} + \frac{x}{xy} = \frac{y + x}{xy} \). This simplifies the numerator of the complex fraction.
03

Combine the numerator and the denominator.

Now the complex fraction is \(\frac{\frac{y + x}{xy}}{x + y} \). To simplify this, remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, rewrite the division as a multiplication: \(\frac{y + x}{xy} \cdot \frac{1}{x + y} \).
04

Simplify and reduce to lowest terms.

In this step you notice that \( y + x \) is the same as \( x + y \). Therefore, this term cancels out in the numerator of the first fraction and the denominator of the second, yielding 1. The result is \(\frac{1}{xy} \). This is the simplest form of the given complex rational expression.

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