To find the least common multiple (LCM) of the denominators of the fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{5}{6}\)[/tex], and [tex]\(\frac{9}{4}\)[/tex], follow these steps:
1. List the denominators: 3, 6, and 4.
2. Find the prime factorization of each denominator:
- [tex]\(3\)[/tex] is already a prime number.
- [tex]\(6\)[/tex] can be expressed as [tex]\(2 \times 3\)[/tex].
- [tex]\(4\)[/tex] can be expressed as [tex]\(2^2\)[/tex].
3. Identify the highest power of each prime that appears in the factorizations:
- For the prime number [tex]\(2\)[/tex], the highest power is [tex]\(2^2\)[/tex] (from 4).
- For the prime number [tex]\(3\)[/tex], the highest power is [tex]\(3\)[/tex] (appearing in both 3 and 6).
4. Multiply these highest powers together to get the LCM:
[tex]\[
\text{LCM} = 2^2 \times 3 = 4 \times 3 = 12
\][/tex]
Therefore, the least common multiple (LCM) of the denominators 3, 6, and 4 is [tex]\(12\)[/tex].
Hence, the best answer is:
B. 12
