Problem 11 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{8}{y^{2}-9} \text { and } \frac{14}{y(y+3)}$$

Step-by-Step Solution

Answer

The least common denominator of the rational expressions is \(y(y + 3)(y - 3)\).

Step 1: Factoring the Denominators

Firstly, we factor the denominators. \(y^2 - 9\) factors into \((y+3)(y-3)\) using the difference of squares method. The second denominator, \(y(y+3)\), is already factored into \(y\) and \(y+3\).

Step 2: Find the LCM of the Factors

From factoring, we have the factors \(y\), \(y + 3\), and \(y - 3\). We then identify the least common multiple (LCM) of these factors. The LCM must include each factor the maximum number of times it appears in either factorization. Hence, the LCM of these factors (and thus, the LCD of the fractions) is \(y(y + 3)(y - 3)\).