To find 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 determine the least common multiple (LCM) of their denominators.
The denominators are:
- 3
- 4
- 32
- 9
Let's break down the process to find the LCM step-by-step:
1. Prime Factorization:
- 3 is a prime number, so its prime factorization is [tex]\(3^1\)[/tex].
- 4 can be factored into [tex]\(2^2\)[/tex].
- 32 can be factored into [tex]\(2^5\)[/tex].
- 9 can be factored into [tex]\(3^2\)[/tex].
2. List out all prime factors:
- For the number 2: [tex]\(2^5\)[/tex] (from the factorization of 32).
- For the number 3: [tex]\(3^2\)[/tex] (from the factorization of 9).
3. Take the highest power of each prime number:
- For 2, the highest power seen is [tex]\(2^5\)[/tex].
- For 3, the highest power seen is [tex]\(3^2\)[/tex].
4. Compute the LCM by multiplying these highest powers together:
[tex]\[
LCM = 2^5 \times 3^2
\][/tex]
5. Calculate the numerical value:
[tex]\[
2^5 = 32
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
32 \times 9 = 288
\][/tex]
So, the 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]\(288\)[/tex].
Therefore, the correct answer is:
C. 288
