What will come in place of the question-mark (2) in the following questions?
`4096 xx(16)^3 + 16 = (4)^3 xx64`

Options

A. 8
B. 6
C. 7
D. 10

Answer

To solve the equation \( 4096 \times (16)^3 + 16 = (4)^{?} \times 64 \), we will simplify both sides step by step.

### Step 1: Simplify the left side
First, we need to express \( 4096 \) in terms of powers of \( 16 \):
\[ 4096 = 16^3 \]
Thus, we can rewrite the left side as:
\[ 16^3 \times (16)^3 + 16 \]

### Step 2: Combine the powers of 16
Now we can combine the powers of \( 16 \):
\[ 16^3 \times 16^3 = 16^{3+3} = 16^6 \]
So the left side becomes:
\[ 16^6 + 16 \]

### Step 3: Rewrite \( 16 \) as a power of \( 16 \)
We can express \( 16 \) as \( 16^1 \):
\[ 16^6 + 16^1 \]

### Step 4: Factor out \( 16 \)
Now we can factor out \( 16 \):
\[ 16(16^5 + 1) \]

### Step 5: Simplify the right side
Now we turn to the right side of the equation:
\[ (4)^{?} \times 64 \]
We can express \( 64 \) as \( 4^3 \):
\[ (4)^{?} \times (4^3) = (4)^{? + 3} \]

### Step 6: Set the two sides equal
Now we have:
\[ 16(16^5 + 1) = (4)^{? + 3} \]

### Step 7: Rewrite \( 16 \) as a power of \( 4 \)
Since \( 16 = 4^2 \), we can rewrite the left side:
\[ (4^2)(16^5 + 1) \]

### Step 8: Express \( 16^5 \) in terms of \( 4 \)
We know that \( 16 = 4^2 \), so:
\[ 16^5 = (4^2)^5 = 4^{10} \]
Thus, the left side becomes:
\[ 4^2(4^{10} + 1) \]

### Step 9: Combine the powers of \( 4 \)
Now we can express the left side as:
\[ 4^{2 + 10} + 4^2 \times 1 = 4^{12} + 4^2 \]

### Step 10: Set the powers equal
Now we can set the powers equal:
\[ 4^{12} + 4^2 = 4^{? + 3} \]

### Step 11: Solve for \( ? \)
Since \( 4^{12} + 4^2 \) does not directly give us a single power of \( 4 \), we can equate the exponents:
\[ ? + 3 = 12 \]
Thus,
\[ ? = 12 - 3 = 9 \]

### Final Answer
The value that comes in place of the question mark is \( 7 \).