Question
The LCD for the fractions 1/3, 3/4, 5/32, and 8/9 is
Question 4 options:
A)
288.
B)
24.
C)
64.
D)
3,072.
Question 4 options:
A)
288.
B)
24.
C)
64.
D)
3,072.
Answers
GPT-4o mini
To find the least common denominator (LCD) for the fractions \( \frac{1}{3} \), \( \frac{3}{4} \), \( \frac{5}{32} \), and \( \frac{8}{9} \), we first need to determine the denominators: 3, 4, 32, and 9.
Next, we factor each denominator into its prime factors:
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 32 = 2^5 \)
- \( 9 = 3^2 \)
Now, we take the highest power of each prime factor that appears in these factorizations:
- For the prime \( 2 \), the highest power is \( 2^5 \) (from 32).
- For the prime \( 3 \), the highest power is \( 3^2 \) (from 9).
Now, we calculate the LCD by multiplying these highest powers together:
\[
\text{LCD} = 2^5 \times 3^2 = 32 \times 9 = 288.
\]
Thus, the least common denominator for the given fractions is \(\textbf{288}\).
The correct answer is **A) 288**.
Next, we factor each denominator into its prime factors:
- \( 3 = 3^1 \)
- \( 4 = 2^2 \)
- \( 32 = 2^5 \)
- \( 9 = 3^2 \)
Now, we take the highest power of each prime factor that appears in these factorizations:
- For the prime \( 2 \), the highest power is \( 2^5 \) (from 32).
- For the prime \( 3 \), the highest power is \( 3^2 \) (from 9).
Now, we calculate the LCD by multiplying these highest powers together:
\[
\text{LCD} = 2^5 \times 3^2 = 32 \times 9 = 288.
\]
Thus, the least common denominator for the given fractions is \(\textbf{288}\).
The correct answer is **A) 288**.