Problem 14 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{9}{y^{2}-25} \text { and } \frac{y}{y^{2}-10 y+25}$$

Step-by-Step Solution

Answer

The least common denominator of \( \frac{9}{y^{2}-25} \) and \( \frac{y}{y^{2}-10 y+25} \) is \( (y-5)^2(y+5) \).

Step 1: Factorize the denominators

Factorize each of the denominators. The factorization of \( y^{2}-25 \) is \( (y-5)(y+5) \). The factorization of \( y^{2}-10y+25 \) is \( (y-5)^2 \).

Step 2: Identify the common factors and additional factors

Looking at the factors, we can see that \( y-5 \) is a common factor. The other factors are \( y+5 \) from the first expression and another \( y-5 \) from the second expression.

Step 3: Assemble the least common denominator

The least common denominator is found by taking the common factor \( y-5 \) and each of the additional factors \( y+5 \) and \( y-5 \). Hence, the LCD is \( (y-5)^2(y+5) \).