Problem 2 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{11}{25 x^{2}} \text { and } \frac{17}{35 x}$$

Step-by-Step Solution

Answer

The least common denominator between the given fractions is \(175x^{2}\).

Step 1: Identify the Denominators

The denominators of the given fractions are \(25x^{2}\) and \(35x\). This you can observe directly from the problem statement.

Step 2: Prime Factorization

In this step you need to find the prime factorizations of each part of your denominators. The prime factorization of 25 is \(5^{2}\), of 35 is \(5 \times 7\), \(x^{2}\) is the prime factorization of the first \(x^{2}\), and for the last \(x\) it's \(x\).

Step 3: Least Common Multiple (LCM)

The LCM is found by multiplying the highest power of all primes numbers gotten from the fractions. That means: \(5^{2}\), 7 and \(x^{2}\) are all included in the LCM. So, the LCM of \(25x^{2}\) and \(35x\) is \(5^{2} \times 7 \times x^{2}\), which simplifies to \(175 x^{2}\).