Problem 63 - College Algebra Essentials

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

Step 1: 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, \( xy \).

Step 2: 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.

Step 3: 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: \(\frac{y + x}{xy} \cdot \frac{1}{x + y} \).

Step 4: 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.