Problem 12 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{14}{y^{2}-49} \text { and } \frac{12}{y(y-7)}$$

Step-by-Step Solution

Answer

The least common denominator of the given rational expressions is \(y^{2}(y^{2} - 49)\).

Step 1: Factoring Denominators

Factoring the denominators of the given rational expressions, we get \(\frac{14}{(y - 7)(y + 7)}\) and \(\frac{12}{y(y - 7)}.\)

Step 2: Determining the Least Common Denominator

The LCD is the least common multiple of the denominators. So, we look for the factors that appear in both expressions. They are \(y\), \(y - 7\), and \(y + 7\). The highest power of each factor in either denominator is the power of that factor in the LCD. Therefore the LCD of the given rational expressions is \(y(y - 7)(y + 7).\)

Step 3: Simplifying the LCD

Finally, simplifying the LCD, we get \(y^{2}(y^{2} - 49)\).