Problem 13 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{7}{y^{2}-1} \text { and } \frac{y}{y^{2}-2 y+1}$$

Step-by-Step Solution

Answer

The least common denominator of \(\frac{7}{y^{2}-1}\) and \(\frac{y}{y^{2}-2 y+1}\) is \((y-1)^{2}(y+1)\).

Step 1: Factorize the denominators

Start by factorizing the denominators of the rational expressions. The denominator \(y^{2}-1\) can be factored as \((y-1)(y+1)\) using the difference of two squares rule. The denominator \(y^{2}-2 y+1\) can be factored as \((y-1)^{2}\), since it represents a perfect square trinomial.

Step 2: Determine the Least Common Multiple (LCM)

Next, find the least common multiple (LCM) of the factored denominators, \((y-1)(y+1)\) and \((y-1)^{2}\). The LCM of two expressions is the product of the highest power of all the factors that appear in either expression. Therefore, the LCD will be \((y-1)^{2}(y+1)\), because it includes both \((y-1)^{2}\) and \(y+1\) which appears only once in the first denominator.