To determine the least common multiple (LCM) of the denominators for the given fractions [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{5}{6}\)[/tex], and [tex]\(\frac{9}{4}\)[/tex], we first need to identify the denominators, which are 3, 6, and 4.
Here is the step-by-step solution:
1. List the denominators: The denominators we need to work with are 3, 6, and 4.
2. Prime factorize each denominator:
- 3 is already a prime number, so the prime factorization of 3 is [tex]\(3\)[/tex].
- 6 can be factorized into [tex]\(6 = 2 \times 3\)[/tex].
- 4 can be factorized into [tex]\(4 = 2 \times 2\)[/tex].
3. Identify the highest power of each prime factor:
- For the prime number 2, the highest power in the factorizations is [tex]\(2^2\)[/tex] (which comes from 4).
- For the prime number 3, the highest power in the factorizations is [tex]\(3\)[/tex] (which comes from both 3 and 6).
4. Multiply these highest powers together to find the LCM:
- The LCM is calculated by multiplying [tex]\(2^2\)[/tex] (from the highest power of 2) and [tex]\(3\)[/tex] (from the highest power of 3):
[tex]\[
\text{LCM} = 2^2 \times 3 = 4 \times 3 = 12
\][/tex]
Therefore, the least common multiple of 3, 6, and 4 is 12.
Hence, the least common denominator for [tex]\(\frac{2}{3}\)[/tex], [tex]\(\frac{5}{6}\)[/tex], and [tex]\(\frac{9}{4}\)[/tex] is [tex]\(\boxed{12}\)[/tex].
