Question

whats the least common denominator for adding fractions \\(\frac{4}{15}\\), \\(\frac{5}{48}\\), and \\(\frac{7}{32}\\)?

Explanation:

Step1: Factorize each denominator

Factorize \( 15 \): \( 15 = 3\times5 \)
Factorize \( 48 \): \( 48 = 2^4\times3 \)
Factorize \( 32 \): \( 32 = 2^5 \)

Step2: Identify the highest power of each prime factor

For prime factor \( 2 \), the highest power is \( 2^5 \) (from \( 32 \)).
For prime factor \( 3 \), the highest power is \( 3^1 \) (from \( 15 \) or \( 48 \)).
For prime factor \( 5 \), the highest power is \( 5^1 \) (from \( 15 \)).

Step3: Calculate the least common denominator (LCD)

Multiply these highest powers together: \( \text{LCD} = 2^5\times3\times5 \)
Calculate \( 2^5 = 32 \), then \( 32\times3 = 96 \), and \( 96\times5 = 480 \).

Answer:

The least common denominator is \( 480 \).