Problem 7 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{5}{7(y+2)} \text { and } \frac{10}{y}$$

Step-by-Step Solution

Answer

The least common denominator (LCD) for the given rational expressions is \(7y(y+2)\).

Step 1: Identifying Denominators

First, identify the denominators in each rational expression. They are \(7(y+2)\) and \(y\).

Step 2: Break Down Denominators into Prime Factors

Identifying the prime factors of a number involves dividing the number by prime numbers, starting from 2, until the number becomes a prime number. For \(7(y+2)\), 7 is a prime number, and \(y+2\) is a binomial which cannot be factored further. For \(y\), it's a single variable and no further factoring is needed.

Step 3: Determining the LCD

The rules of determining the least common denominator state that it must be the product of highest powers of all factors present in the denominators. Here, the factors are 7, \(y\) and \((y+2)\) and they all appear to the power of 1, which is their highest occurring power. Hence, the least common denominator (LCD) of these two expressions is \(7(y)\times(y+2)\).