To find the least common denominator (LCD) of the fractions [tex]\(\frac{1}{3}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], [tex]\(\frac{5}{32}\)[/tex], and [tex]\(\frac{8}{9}\)[/tex], we need to determine the least common multiple (LCM) of their denominators: 3, 4, 32, and 9.
Let's go through the process step-by-step:
1. List the prime factors of each denominator:
- [tex]\(3\)[/tex] is already a prime number.
- [tex]\(4\)[/tex] can be factored as [tex]\(2^2\)[/tex].
- [tex]\(32\)[/tex] can be factored as [tex]\(2^5\)[/tex].
- [tex]\(9\)[/tex] can be factored as [tex]\(3^2\)[/tex].
2. Identify the maximum power of each prime number appearing in the factorizations:
- For the prime number [tex]\(2\)[/tex], the highest power is [tex]\(2^5\)[/tex] (from 32).
- For the prime number [tex]\(3\)[/tex], the highest power is [tex]\(3^2\)[/tex] (from 9).
3. Multiply these highest powers to get the LCM:
[tex]\[
\text{LCM} = 2^5 \times 3^2
\][/tex]
Calculate this step-by-step:
- [tex]\(2^5 = 32\)[/tex]
- [tex]\(3^2 = 9\)[/tex]
- Multiply these values together: [tex]\(32 \times 9 = 288\)[/tex]
Therefore, the least common denominator (LCD) for the fractions [tex]\(\frac{1}{3}\)[/tex], [tex]\(\frac{3}{4}\)[/tex], [tex]\(\frac{5}{32}\)[/tex], and [tex]\(\frac{8}{9}\)[/tex] is:
C. 288
