If `{(2^(4))^(1//2)}^(?)=256`, find the value of '?'.

Options

A. 1
B. 2
C. 4
D. 8

Answer

To solve the equation \({(2^{4})^{(1/2)}}^{?} = 256\), we will follow these steps:

### Step 1: Simplify the left-hand side
We start with the expression \((2^{4})^{(1/2)}\). According to the laws of exponents, when raising a power to another power, we multiply the exponents:

\[ (2^{4})^{(1/2)} = 2^{4 \cdot (1/2)} = 2^{2} \]

### Step 2: Rewrite the equation
Now we can rewrite the original equation using our simplification:

\[ (2^{2})^{?} = 256 \]

### Step 3: Further simplify the left-hand side
Using the same exponent rule, we can simplify the left-hand side further:

\[ 2^{2 \cdot ?} = 256 \]

### Step 4: Express 256 as a power of 2
Next, we need to express 256 as a power of 2. We know that:

\[ 256 = 2^{8} \]

### Step 5: Set the exponents equal
Now that both sides of the equation have the same base (which is 2), we can set the exponents equal to each other:

\[ 2 \cdot ? = 8 \]

### Step 6: Solve for ?
To find the value of \(?\), we divide both sides by 2:

\[ ? = \frac{8}{2} = 4 \]

### Final Answer
Thus, the value of \(?\) is:

\[ \boxed{4} \]

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