Question
What is the least common denominator of the fractions 2/27 , 1/12 AND 5/9
Answers
Answered by
GPT-4o mini
To find the least common denominator (LCD) of the fractions \( \frac{2}{27} \), \( \frac{1}{12} \), and \( \frac{5}{9} \), we first need to determine the denominators: 27, 12, and 9.
Next, we find the prime factorization of each denominator:
- \( 27 = 3^3 \)
- \( 12 = 2^2 \times 3^1 \)
- \( 9 = 3^2 \)
Now, we find the highest power of each prime factor present in any of the factorizations:
- The highest power of 2 is \( 2^2 \) from 12.
- The highest power of 3 is \( 3^3 \) from 27.
Now, we calculate the least common denominator by multiplying these highest powers together:
\[
\text{LCD} = 2^2 \times 3^3 = 4 \times 27
\]
Calculating this gives:
\[
4 \times 27 = 108
\]
Therefore, the least common denominator of the fractions \( \frac{2}{27} \), \( \frac{1}{12} \), and \( \frac{5}{9} \) is \( \boxed{108} \).
Next, we find the prime factorization of each denominator:
- \( 27 = 3^3 \)
- \( 12 = 2^2 \times 3^1 \)
- \( 9 = 3^2 \)
Now, we find the highest power of each prime factor present in any of the factorizations:
- The highest power of 2 is \( 2^2 \) from 12.
- The highest power of 3 is \( 3^3 \) from 27.
Now, we calculate the least common denominator by multiplying these highest powers together:
\[
\text{LCD} = 2^2 \times 3^3 = 4 \times 27
\]
Calculating this gives:
\[
4 \times 27 = 108
\]
Therefore, the least common denominator of the fractions \( \frac{2}{27} \), \( \frac{1}{12} \), and \( \frac{5}{9} \) is \( \boxed{108} \).