To find the least common denominator (LCD) of the fractions \( \frac{5}{9} \) and \( \frac{1}{2} \), we need to determine the least common multiple (LCM) of the denominators, which are 9 and 2.
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Identify the denominators: The denominators are 9 and 2.
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Find the prime factorization:
- The prime factorization of 9 is \( 3^2 \).
- The prime factorization of 2 is \( 2^1 \).
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Determine the LCM:
- For each prime factor, take the highest power that appears in either factorization:
- For prime factor 2: the highest power is \( 2^1 \).
- For prime factor 3: the highest power is \( 3^2 \).
- For each prime factor, take the highest power that appears in either factorization:
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Multiply these together to find the LCM: \[ \text{LCM} = 2^1 \times 3^2 = 2 \times 9 = 18 \]
Thus, the least common denominator of \( \frac{5}{9} \) and \( \frac{1}{2} \) is \( \boxed{18} \).