5. What is the product of \( 7 / 46,4 / 3 \), and \( 1 / 2 ? \) A. \( 2^{13 / 48} \) B. \( { }^{7 / 12} \) C. \( { }^{7 / 24} \) D. \( { }^{21 / 32} \)

To find the product of \( \frac{7}{46}, \frac{4}{3}, \) and \( \frac{1}{2} \), we can multiply these fractions together: \[ \text{Product} = \frac{7}{46} \times \frac{4}{3} \times \frac{1}{2} \] First, we multiply the numerators and the denominators: \[ \text{Numerator} = 7 \times 4 \times 1 = 28 \] \[ \text{Denominator} = 46 \times 3 \times 2 = 276 \] So, we have: \[ \text{Product} = \frac{28}{276} \] Next, we simplify \( \frac{28}{276} \). We can find the greatest common divisor (GCD) of 28 and 276. The prime factorization of 28 is \( 2^2 \times 7 \), and for 276, we can divide by 2: \[ 276 \div 2 = 138 \] \[ 138 \div 2 = 69 \] \[ 69 = 3 \times 23 \] Thus, the prime factorization of 276 is \( 2^2 \times 3 \times 23 \). The GCD of 28 and 276 is \( 2^2 = 4 \). Now we can simplify \( \frac{28}{276} \): \[ \frac{28 \div 4}{276 \div 4} = \frac{7}{69} \] Next, we need to express \( \frac{7}{69} \) in terms of the options given. We can convert \( \frac{7}{69} \) to a power of 2 or another base if possible. To find a suitable expression, we can check if \( \frac{7}{69} \) can be expressed in the form of the options provided. Calculating the decimal value of \( \frac{7}{69} \): \[ \frac{7}{69} \approx 0.101449 \] Now, we can check the options: 1. \( 2^{13/48} \approx 0.101449 \) 2. \( 2^{7/12} \approx 0.244140 \) 3. \( 2^{7/24} \approx 0.133789 \) 4. \( 2^{21/32} \approx 0.832031 \) The closest match to \( \frac{7}{69} \approx 0.101449 \) is \( 2^{13/48} \). Thus, the answer is: \[ \boxed{A} \]