Problem 1 - Introductory Algebra for College Students

Question

Find the least common denominator of the rational expressions. $$\frac{7}{15 x^{2}} \text { and } \frac{13}{24 x}$$

Step-by-Step Solution

Answer

The least common denominator of the rational expressions \( \frac{7}{15 x^{2}} \) and \( \frac{13}{24 x} \) is \( 120x^2 \).

Step 1: Prime Factorization of the Denominators

Find the prime factorization of each denominator. - For 15, the prime factors are \( 3 \times 5 \)- For \(x^{2}\), the prime factors are \( x \times x \)- For 24, the prime factors are \( 2 \times 2 \times 2 \times 3 \)- For \( x \), the prime factor is \( x \)

Step 2: Identify the Highest Powers of the Prime Factors

Identify the highest power of each prime factor from both denominators. - Highest power of 2: \(2^3\)- Highest power of 3: \(3^1\)- Highest power of 5: \(5^1\)- Highest power of \(x\): \(x^2\)

Step 3: Calculate the Least Common Denominator

Calculate the LCD by multiplying the highest powers of all the prime factors. - LCD = \(2^3 \times 3^1 \times 5^1 \times x^2 = 120x^2\).