Problem 3 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{8}{15 x^{2}} \text { and } \frac{5}{6 x^{5}}$$

Step-by-Step Solution

Answer

The least common denominator of \(\frac{8}{15x^{2}}\) and \(\frac{5}{6x^{5}}\) is \(30x^{5}\).

Step 1 - Factorization

Start by factorizing the denominators of the fractions. Additionally, it can be observed that 15 and 6 (the constants in the denominators) are both divisible by 3, and \(x^{5}\) contains \(x^{2}\): \(15x^{2} = 3 \cdot 5 \cdot x^{2}\) \(6x^{5} = 3 \cdot 2 \cdot x^{2} \cdot x^{3}\)

Step 2 - Identify Common and Individual Factors

Identify the common factors and the factors that are individual to each denominator: Common factors: 3, \(x^{2}\) Individual Factors: 5, 2, \(x^{3}\)

Step 3 - Compute the Least Common Denominator

The LCD is the product of all the common factors and the individual factors of the denominators. Therefore: \(LCD = 3 \cdot 5 \cdot 2 \cdot x^{2} \cdot x^{3} = 30x^{5}\)