If `A=2^(7) times 3^(5) and B = 3^(5) times 2^(3)`, then what is the value of `A times B`?

Options

A. `2^(10) times 3^(10)`
B. `2^(12) times 3^(8)`
C. `6^(12) times 6^(8)`
D. `2^(21) times 3^(35)`

Answer

To find the value of \( A \times B \), we start with the definitions of \( A \) and \( B \):

1. **Given:**
- \( A = 2^7 \times 3^5 \)
- \( B = 3^5 \times 2^3 \)

2. **Multiply \( A \) and \( B \):**
\[ A \times B = (2^7 \times 3^5) \times (3^5 \times 2^3) \]

3. **Rearranging the terms:**
\[ A \times B = 2^7 \times 2^3 \times 3^5 \times 3^5 \]

4. **Using the property of exponents:**
- When multiplying like bases, we add the exponents.
\[ 2^7 \times 2^3 = 2^{7+3} = 2^{10} \]
\[ 3^5 \times 3^5 = 3^{5+5} = 3^{10} \]

5. **Combining the results:**
\[ A \times B = 2^{10} \times 3^{10} \]

6. **Factoring out the common exponent:**
\[ A \times B = (2 \times 3)^{10} = 6^{10} \]

Thus, the value of \( A \times B \) is \( 6^{10} \).