`(4)^(0.5)xx(0.5)^(4)` is equal to :

Options

A. A) `1`
B. B) `4`
C. C) `(1)/(8)`
D. D) `(1)/(32)`

Answer

To solve the expression \((4)^{0.5} \times (0.5)^{4}\), we can follow these steps:

### Step 1: Rewrite the bases in terms of powers of 2
We know that:
\[ 4 = 2^2 \quad \text{and} \quad 0.5 = \frac{1}{2} = 2^{-1} \]
So we can rewrite the expression as:
\[ (2^2)^{0.5} \times (2^{-1})^{4} \]

### Step 2: Apply the power of a power property
Using the property \((a^m)^n = a^{m \cdot n}\), we can simplify each term:
\[ (2^2)^{0.5} = 2^{2 \cdot 0.5} = 2^{1} = 2 \]
\[ (2^{-1})^{4} = 2^{-1 \cdot 4} = 2^{-4} \]

### Step 3: Combine the powers of 2
Now we can combine the two results:
\[ 2 \times 2^{-4} = 2^{1} \times 2^{-4} = 2^{1 - 4} = 2^{-3} \]

### Step 4: Convert the result back to a fraction
The expression \(2^{-3}\) can be rewritten as:
\[ 2^{-3} = \frac{1}{2^{3}} = \frac{1}{8} \]

### Final Answer
Thus, the value of \((4)^{0.5} \times (0.5)^{4}\) is:
\[ \frac{1}{8} \]