To determine 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], we need to find the Least Common Multiple (LCM) of their denominators: 3, 4, 32, and 9.
Step-by-step solution:
1. Identify the denominators:
- The denominators are: 3, 4, 32, and 9.
2. Find the prime factorization of each denominator:
- [tex]\(3\)[/tex] is already a prime number.
- [tex]\(4 = 2^2\)[/tex]
- [tex]\(32 = 2^5\)[/tex]
- [tex]\(9 = 3^2\)[/tex]
3. Determine the maximum power of each prime factor present in these factorizations:
- The prime [tex]\(2\)[/tex] appears with the highest power in [tex]\(32\)[/tex], which is [tex]\(2^5\)[/tex].
- The prime [tex]\(3\)[/tex] appears with the highest power in [tex]\(9\)[/tex], which is [tex]\(3^2\)[/tex].
4. Multiply these maximum powers together to find the LCM:
The LCM will be [tex]\(2^5 \times 3^2\)[/tex].
Let's calculate this:
[tex]\[
2^5 = 32
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
LCM = 32 \times 9 = 288
\][/tex]
Thus, 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 [tex]\(\boxed{288}\)[/tex].
So, the correct answer is:
B. 288
