The value of `[[(2401)^(-1//2)]^(-1//4)]^(2)` is

Options

A. 8
B. 7
C. `(1)/(7)`
D. 16

Answer

To solve the expression \([[(2401)^{-1/2}]^{-1/4}]^{2}\), we will follow these steps:

### Step 1: Simplify the Inner Expression
Start with the inner expression:
\[ (2401)^{-1/2} \]
This means we are taking the reciprocal of the square root of 2401.

### Step 2: Calculate the Square Root of 2401
Next, we need to find the square root of 2401. We know that:
\[ \sqrt{2401} = 49 \quad \text{(since } 49 \times 49 = 2401\text{)} \]
Thus,
\[ (2401)^{-1/2} = \frac{1}{\sqrt{2401}} = \frac{1}{49} \]

### Step 3: Raise to the Power of \(-1/4\)
Now we take the result from Step 2 and raise it to the power of \(-1/4\):
\[ \left(\frac{1}{49}\right)^{-1/4} \]
This means we are taking the reciprocal of the fourth root of 49.

### Step 4: Calculate the Fourth Root of 49
The fourth root of 49 can be simplified as:
\[ \sqrt[4]{49} = \sqrt{\sqrt{49}} = \sqrt{7} \quad \text{(since } \sqrt{49} = 7\text{)} \]
Thus,
\[ \left(\frac{1}{49}\right)^{-1/4} = \frac{1}{\sqrt[4]{49}} = \frac{1}{\sqrt{7}} \]

### Step 5: Raise to the Power of 2
Now, we raise the result from Step 4 to the power of 2:
\[ \left(\frac{1}{\sqrt{7}}\right)^{2} = \frac{1}{7} \]

### Final Answer
Thus, the value of the original expression \([[(2401)^{-1/2}]^{-1/4}]^{2}\) is:
\[ \frac{1}{7} \]