If `4sqrt(3sqrt(x^(2)))=x^(k),` then k=

Options

A. `2/6`
B. 6
C. `1/6`
D. 7

Answer

To solve the equation \( 4\sqrt{3\sqrt{x^2}} = x^k \), we will simplify the left-hand side and find the value of \( k \).

### Step-by-Step Solution:

1. **Simplify the Left Side**:
\[ 4\sqrt{3\sqrt{x^2}} = 4\sqrt{3 \cdot x} \quad \text{(since } \sqrt{x^2} = x \text{)} \]

2. **Rewrite the Square Root**:
\[ 4\sqrt{3x} = 4(3x)^{1/2} \]

3. **Express in Exponential Form**:
\[ 4(3x)^{1/2} = 4 \cdot 3^{1/2} \cdot x^{1/2} \]

4. **Combine the Constants**:
\[ = 4 \cdot \sqrt{3} \cdot x^{1/2} \]

5. **Set Equal to Right Side**:
Now we have:
\[ 4\sqrt{3} \cdot x^{1/2} = x^k \]

6. **Equate the Exponents**:
Since the bases are the same, we can equate the exponents:
\[ \frac{1}{2} = k \]

7. **Final Result**:
Thus, the value of \( k \) is:
\[ k = \frac{1}{2} \]

### Conclusion:
The value of \( k \) is \( \frac{1}{2} \).