To find the Least Common Denominator (LCD) of the fractions [tex]\(\frac{5}{8}\)[/tex], [tex]\(\frac{1}{2}\)[/tex], and [tex]\(\frac{5}{6}\)[/tex], we follow these steps:
1. Identify the Denominators:
- The denominators of the fractions are 8, 2, and 6.
2. List the Denominators:
- We have 8, 2, and 6.
3. Find the Least Common Multiple (LCM) of the Denominators:
- To find the LCM of the numbers 8, 2, and 6, we need to identify the smallest number that all three denominators can divide without leaving a remainder.
4. Prime Factorization:
- Prime factorize each denominator:
- [tex]\(8 = 2^3\)[/tex]
- [tex]\(2 = 2^1\)[/tex]
- [tex]\(6 = 2^1 \times 3^1\)[/tex]
5. Combine the Prime Factors Taking the Highest Power of Each Factor:
- For the factor 2, the highest power is [tex]\(2^3\)[/tex] (from 8).
- For the factor 3, the highest power is [tex]\(3^1\)[/tex] (from 6).
- Multiply these together to find the LCM:
[tex]\[
\text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
\][/tex]
6. Conclude the LCD:
- The Least Common Denominator of [tex]\(\frac{5}{8}\)[/tex], [tex]\(\frac{1}{2}\)[/tex], and [tex]\(\frac{5}{6}\)[/tex] is 24.
Therefore, the Least Common Denominator (LCD) of the fractions [tex]\(\frac{5}{8}\)[/tex], [tex]\(\frac{1}{2}\)[/tex], and [tex]\(\frac{5}{6}\)[/tex] is [tex]\(\boxed{24}\)[/tex].
