To determine the least common denominator (LCD) of the fractions [tex]\( \frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \frac{8}{9} \)[/tex], follow these steps:
1. Identify the denominators: The denominators of the fractions are 3, 4, 32, and 9.
2. List the prime factors:
- 3 is a prime number.
- [tex]\( 4 = 2^2 \)[/tex]
- [tex]\( 32 = 2^5 \)[/tex]
- [tex]\( 9 = 3^2 \)[/tex]
3. Find the highest powers of all prime factors:
- The prime factor 2 appears in the denominators as [tex]\( 2^5 \)[/tex] (from 32), which is the highest power of 2.
- The prime factor 3 appears as [tex]\( 3^2 \)[/tex] (from 9), which is the highest power of 3.
4. Calculate the LCM (LCD):
- Multiply the highest powers of all prime factors together:
[tex]\[
\text{LCM} = 2^5 \times 3^2
\][/tex]
- Calculate each part:
[tex]\[
2^5 = 32
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
32 \times 9 = 288
\][/tex]
Therefore, the least common denominator (LCD) of the fractions [tex]\( \frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \frac{8}{9} \)[/tex] is 288.
Thus, the best answer is:
A. 288
