If `A = 4^(5) xx 7^(3) and B = 7^(2) xx 4^(6)`, then what is the value of `A xx B` ?

Options

A. `4^(11) xx 7^(5)`
B. `4^(10) xx 7^(18)`
C. `4^(30) xx 7^(6)`
D. `4^(7) xx 7^(9)`

Answer

To find the value of \( A \times B \) where \( A = 4^5 \times 7^3 \) and \( B = 7^2 \times 4^6 \), we can follow these steps:

### Step 1: Write down the expressions for A and B
We have:
\[ A = 4^5 \times 7^3 \]
\[ B = 7^2 \times 4^6 \]

### Step 2: Combine the expressions for A and B
To find \( A \times B \), we can multiply the two expressions:
\[ A \times B = (4^5 \times 7^3) \times (7^2 \times 4^6) \]

### Step 3: Rearrange the multiplication
We can rearrange the multiplication to group the same bases together:
\[ A \times B = 4^5 \times 4^6 \times 7^3 \times 7^2 \]

### Step 4: Apply the exponent multiplication rule
Using the property of exponents that states \( a^m \times a^n = a^{m+n} \), we can simplify:
\[ A \times B = 4^{5+6} \times 7^{3+2} \]

### Step 5: Calculate the new exponents
Now we can calculate the new exponents:
\[ A \times B = 4^{11} \times 7^{5} \]

### Final Result
Thus, the value of \( A \times B \) is:
\[ A \times B = 4^{11} \times 7^{5} \]