What should come in place of the question mark (?) in the following questions ?
`(64)^(3) xx (8)^(2) xx (512)^(2) = (8)^(?)`

Options

A. 8
B. 4
C. 14
D. 10

Answer

To solve the equation \( (64)^3 \times (8)^2 \times (512)^2 = (8)^? \), we will first express all the numbers in terms of base 8.

### Step 1: Convert each term to base 8
- \( 64 = 8^2 \)
- \( 8 = 8^1 \)
- \( 512 = 8^3 \)

### Step 2: Rewrite the equation using base 8
Now we can rewrite the equation:
\[ (8^2)^3 \times (8^1)^2 \times (8^3)^2 = (8)^? \]

### Step 3: Simplify each term
Using the power of a power property \((a^m)^n = a^{m \cdot n}\):
- \( (8^2)^3 = 8^{2 \cdot 3} = 8^6 \)
- \( (8^1)^2 = 8^{1 \cdot 2} = 8^2 \)
- \( (8^3)^2 = 8^{3 \cdot 2} = 8^6 \)

### Step 4: Combine the terms on the left side
Now we can combine the terms:
\[ 8^6 \times 8^2 \times 8^6 = 8^{6 + 2 + 6} \]
Calculating the exponent:
\[ 6 + 2 + 6 = 14 \]
So, we have:
\[ 8^{14} = (8)^? \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ ? = 14 \]

### Final Answer
Thus, the value that should come in place of the question mark (?) is \( 14 \).

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