In each of the following questions, find the exact value of (?)
`2^(3) xx 4^(6) div 8^(2) = (2)^(? - 2)`

Options

A. 7
B. 6
C. 9
D. 11

Answer

To solve the equation \( 2^3 \times 4^6 \div 8^2 = 2^{? - 2} \), we will follow these steps:

### Step 1: Rewrite all terms with base 2
First, we need to express all terms in the equation using the same base, which is 2.

- \( 4 \) can be rewritten as \( 2^2 \), so \( 4^6 = (2^2)^6 = 2^{12} \).
- \( 8 \) can be rewritten as \( 2^3 \), so \( 8^2 = (2^3)^2 = 2^6 \).

Now, substituting these into the equation gives us:
\[ 2^3 \times 2^{12} \div 2^6 \]

### Step 2: Apply the properties of exponents
Using the properties of exponents, we can simplify the left side of the equation.

1. Multiply the terms:
\[ 2^3 \times 2^{12} = 2^{3 + 12} = 2^{15} \]

2. Now divide by \( 2^6 \):
\[ 2^{15} \div 2^6 = 2^{15 - 6} = 2^9 \]

So, the left side simplifies to \( 2^9 \).

### Step 3: Set the equation
Now we have:
\[ 2^9 = 2^{? - 2} \]

### Step 4: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ 9 = ? - 2 \]

### Step 5: Solve for ?
To find the value of \( ? \), we add 2 to both sides:
\[ ? = 9 + 2 = 11 \]

Thus, the exact value of \( ? \) is \( 11 \).

### Final Answer:
The value of \( ? \) is \( 11 \).

---