Problem 4 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{7}{15 x^{2}} \text { and } \frac{11}{24 x^{5}}$$

Step-by-Step Solution

Answer

The least common denominator of \(\frac{7}{15 x^{2}}\) and \(\frac{11}{24 x^{5}}\) is \(120x^5\).

Step 1: Factorize Denominators

In order to find the common denominators we need to factorize each of the denominators: We start by \(15x^2\), here the prime factorization is \(3 * 5 * x * x\). We then prime factorize the second denominator \(24x^5 = 2 * 2 * 2 * 3 * x * x * x * x * x\).

Step 2: Select Maximum Powers

The next step is to take maximum power of each prime factors present in the denominators of given expressions i.e. \(2^3, 3^1, 5^1, x^5\). Only the maximum exponent for each prime number is selected.

Step 3: Calculate the Least Common Denominator (LCD)

Multiply together each unique factor, to the power of the highest exponent that factor had in either factorization. So LCD = 2^3 * 3^1 * 5^1 * x^5 = 120x^5.