To determine the Least Common Denominator (LCD) for the fractions [tex]\( \frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \text{ and } \frac{8}{9} \)[/tex], we need to find the least common multiple (LCM) of their denominators: 3, 4, 32, and 9.
Here is a step-by-step method to find the LCD:
1. List the denominators: The denominators are 3, 4, 32, and 9.
2. Determine the prime factorization of each denominator:
- [tex]\(3\)[/tex] is already a prime number.
- [tex]\(4\)[/tex] can be factorized into [tex]\(2^2\)[/tex].
- [tex]\(32\)[/tex] can be factorized into [tex]\(2^5\)[/tex].
- [tex]\(9\)[/tex] can be factorized into [tex]\(3^2\)[/tex].
3. Identify the highest powers of all prime numbers appearing in these factorizations:
- The prime number [tex]\(2\)[/tex] appears in [tex]\(2^2\)[/tex] and [tex]\(2^5\)[/tex]. The highest power is [tex]\(2^5\)[/tex].
- The prime number [tex]\(3\)[/tex] appears in [tex]\(3\)[/tex] and [tex]\(3^2\)[/tex]. The highest power is [tex]\(3^2\)[/tex].
4. Multiply these highest powers together to find the LCM:
[tex]\[
2^5 \times 3^2 = 32 \times 9
\][/tex]
[tex]\[
32 \times 9 = 288
\][/tex]
Therefore, the LCD for the fractions [tex]\( \frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \text{ and } \frac{8}{9} \)[/tex] is [tex]\( 288 \)[/tex].
So, the correct answer is:
C. 288.
