Simplify : `[64^((2)/(3))xx2^(-2)xx8^(0)]^((1)/(2))`

Options

A. a) `0`
B. b) `1`
C. c) `2`
D. d) `(1)/(2)`

Answer

To simplify the expression \([64^{(2/3)} \times 2^{-2} \times 8^{0}]^{(1/2)}\), we will follow these steps:

### Step 1: Simplify each component inside the brackets
1. **Calculate \(64^{(2/3)}\)**:
- We know that \(64 = 4^3\), so we can rewrite it as:
\[ 64^{(2/3)} = (4^3)^{(2/3)} = 4^{(3 \cdot \frac{2}{3})} = 4^2 = 16 \]

2. **Calculate \(2^{-2}\)**:
- This is simply:
\[ 2^{-2} = \frac{1}{2^2} = \frac{1}{4} \]

3. **Calculate \(8^{0}\)**:
- Any number raised to the power of 0 is 1:
\[ 8^{0} = 1 \]

### Step 2: Combine the results
Now we can combine the results from Step 1:
\[ 64^{(2/3)} \times 2^{-2} \times 8^{0} = 16 \times \frac{1}{4} \times 1 \]

### Step 3: Simplify the multiplication
1. **Calculate \(16 \times \frac{1}{4}\)**:
\[ 16 \times \frac{1}{4} = \frac{16}{4} = 4 \]

### Step 4: Apply the outer exponent
Now we apply the outer exponent \((1/2)\):
\[ [4]^{(1/2)} = \sqrt{4} = 2 \]

### Final Answer
Thus, the simplified value of the expression \([64^{(2/3)} \times 2^{-2} \times 8^{0}]^{(1/2)}\) is:
\[ \boxed{2} \]