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

Options

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

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ (4)^{0.5} \times (0.5)^{4} \]

### Step 2: Express 4 as a power of 2
We know that \( 4 = 2^2 \). Therefore, we can rewrite \( 4^{0.5} \) as:
\[ (2^2)^{0.5} \]

### Step 3: Apply the power of a power rule
Using the power of a power rule \( (a^m)^n = a^{m \cdot n} \), we get:
\[ (2^2)^{0.5} = 2^{2 \cdot 0.5} = 2^{1} = 2 \]

### Step 4: Rewrite \( 0.5 \) as a fraction
Next, we rewrite \( 0.5 \) as \( \frac{1}{2} \). Thus, we have:
\[ (0.5)^{4} = \left(\frac{1}{2}\right)^{4} \]

### Step 5: Apply the power of a fraction rule
Using the rule \( \left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}} \), we can write:
\[ \left(\frac{1}{2}\right)^{4} = \frac{1^{4}}{2^{4}} = \frac{1}{16} \]

### Step 6: Combine the results
Now we can combine the results from Steps 3 and 5:
\[ 2 \times \frac{1}{16} = \frac{2}{16} \]

### Step 7: Simplify the fraction
Finally, we simplify \( \frac{2}{16} \):
\[ \frac{2}{16} = \frac{1}{8} \]

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

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