What value will come in place of question mark (?) in the questions given below?
`(64)^(2x) xx (128)^(-x) + (32)^(-x/2) = (2)^(? + (x - 1/2))`

Options

A. `(x)/(2) - 1`
B. `(3x)/(2) + 1`
C. `(x + 1)/(2)`
D. `(3x + 1)/(2)`

Answer

To solve the equation \( (64)^{2x} \cdot (128)^{-x} + (32)^{-x/2} = (2)^{? + (x - 1/2)} \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
- \( 64 = 2^6 \)
- \( 128 = 2^7 \)
- \( 32 = 2^5 \)

Thus, we can rewrite the equation as:
\[ (2^6)^{2x} \cdot (2^7)^{-x} + (2^5)^{-x/2} = (2)^{? + (x - 1/2)} \]

### Step 2: Simplify the left-hand side
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we simplify:
\[ 2^{12x} \cdot 2^{-7x} + 2^{-5x/2} = 2^{? + (x - 1/2)} \]

Now, combine the exponents on the left:
\[ 2^{12x - 7x} + 2^{-5x/2} = 2^{? + (x - 1/2)} \]
This simplifies to:
\[ 2^{5x} + 2^{-5x/2} = 2^{? + (x - 1/2)} \]

### Step 3: Express the left-hand side with a common base
We can express \(2^{5x}\) and \(2^{-5x/2}\) with a common base:
\[ 2^{5x} + 2^{-5x/2} = 2^{5x} + \frac{1}{2^{5x/2}} = 2^{5x} + 2^{-5x/2} \]

### Step 4: Find a common exponent
To combine these terms, we can express \(2^{5x}\) in terms of \(2^{-5x/2}\):
\[ 2^{5x} = 2^{5x/2} \cdot 2^{5x/2} = 2^{5x/2} \cdot 2^{5x/2} + 2^{-5x/2} \]

### Step 5: Set the equation
Now we can set the equation:
\[ 2^{5x/2} \cdot 2^{5x/2} + 2^{-5x/2} = 2^{? + (x - 1/2)} \]

### Step 6: Solve for the exponent
Now we can equate the exponents:
\[ 5x + \left(-\frac{5x}{2}\right) = ? + x - \frac{1}{2} \]

### Step 7: Combine like terms
This gives us:
\[ \frac{10x}{2} - \frac{5x}{2} = ? + x - \frac{1}{2} \]
\[ \frac{5x}{2} = ? + x - \frac{1}{2} \]

### Step 8: Isolate the question mark
Rearranging gives:
\[ ? = \frac{5x}{2} - x + \frac{1}{2} \]
\[ ? = \frac{5x - 2x + 1}{2} \]
\[ ? = \frac{3x + 1}{2} \]

### Final Answer
Thus, the value that will come in place of the question mark (?) is:
\[ \frac{3x + 1}{2} \]