`{4sqrt((1/x)^(-12))}^(-2//3)=`

Options

A. `1/x^(2)`
B. `1/x^(4)`
C. `1/x^(3)`
D. `1/x`

Answer

To solve the expression \({4\sqrt{(1/x)^{-12}}}^{(-2/3)}\), we will simplify it step by step.

### Step 1: Rewrite the expression
The expression can be rewritten using the property of roots:
\[ 4\sqrt{(1/x)^{-12}} = (1/x)^{-12} \text{ raised to the power of } \frac{1}{4} \]
Thus, we have:
\[ (1/x)^{-12} = \left((1/x)^{-12}\right)^{\frac{1}{4}} \]

### Step 2: Apply the exponent
Now, we apply the exponent \(-\frac{2}{3}\) to the entire expression:
\[ \left((1/x)^{-12}\right)^{\frac{1}{4}} = (1/x)^{-12 \cdot \frac{1}{4}} = (1/x)^{-3} \]
So, we can rewrite the expression as:
\[ (1/x)^{-3} = (1/x)^{3} \]

### Step 3: Simplify the expression
Now, we can simplify \((1/x)^{3}\):
\[ (1/x)^{3} = \frac{1}{x^3} \]

### Final Result
Thus, the final simplified expression is:
\[ \frac{1}{x^3} \]