To find the Least Common Denominator (LCD) for the given fractions [tex]\( \frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \)[/tex] and [tex]\( \frac{8}{9} \)[/tex], the following steps can be followed:
1. Identify the denominators of each fraction:
[tex]\[3, 4, 32, 9\][/tex]
2. Find the least common multiple (LCM) of these denominators, which is the smallest number that each denominator can divide into without leaving a remainder.
- The denominators are: [tex]\(3\)[/tex], [tex]\(4\)[/tex], [tex]\(32\)[/tex], and [tex]\(9\)[/tex].
- We need to find a common multiple of these numbers.
3. Calculate the LCM step by step:
- Prime factorize each denominator:
[tex]\[
\begin{align*}
3 &= 3^1 \\
4 &= 2^2 \\
32 &= 2^5 \\
9 &= 3^2
\end{align*}
\][/tex]
- Determine the highest power of each prime number present in the factorization:
[tex]\[
\begin{align*}
2: & \ \ \ 2^5 \text{ (from 32)} \\
3: & \ \ \ 3^2 \text{ (from 9)}
\end{align*}
\][/tex]
- Multiply these together to find the LCM:
[tex]\[
\mathrm{LCM} = 2^5 \times 3^2 = 32 \times 9 = 288
\][/tex]
Therefore, the least common denominator (LCD) for the fractions [tex]\( \frac{1}{3}, \frac{3}{4}, \frac{5}{32}, \)[/tex] and [tex]\( \frac{8}{9} \)[/tex] is [tex]\( 288 \)[/tex].
So, the correct answer is:
C. [tex]\(288\)[/tex]
