What should come in place of question mark (?) in following questions ?
`(4)^(?) xx 2 = ((16)^(2))/(root(4)(16))`

Options

A. 2
B. 3
C. 4
D. 1

Answer

To solve the equation \( (4)^{?} \times 2 = \frac{(16)^{2}}{\sqrt[4]{16}} \), we will follow these steps:

### Step 1: Simplify the Right Side of the Equation
First, we need to simplify the right side of the equation, which is \( \frac{(16)^{2}}{\sqrt[4]{16}} \).

1. Calculate \( (16)^{2} \):
\[ (16)^{2} = 256 \]

2. Calculate \( \sqrt[4]{16} \):
- We know that \( 16 = 4^{2} \), so:
\[ \sqrt[4]{16} = \sqrt[4]{4^{2}} = 4^{\frac{2}{4}} = 4^{\frac{1}{2}} = 2 \]

3. Now substitute these values back into the equation:
\[ \frac{256}{2} = 128 \]

### Step 2: Set Up the Equation
Now we have:
\[ (4)^{?} \times 2 = 128 \]

### Step 3: Isolate \( (4)^{?} \)
To isolate \( (4)^{?} \), divide both sides by 2:
\[ (4)^{?} = \frac{128}{2} = 64 \]

### Step 4: Rewrite 64 in Terms of Base 4
Next, we need to express 64 as a power of 4:
\[ 64 = 4^{3} \quad \text{(since \( 4^{3} = 64 \))} \]

### Step 5: Equate the Exponents
Now we can equate the exponents:
\[ ? = 3 \]

### Conclusion
Thus, the value that should come in place of the question mark (?) is **3**.