`root4(root3(2^(2)))` equal to

Options

A. `2^(-1/6)`
B. `2^(-6)`
C. `2^(1/6)`
D. `2^(6)`

Explanation

To solve the expression \( \sqrt[4]{\sqrt[3]{2^2}} \), we will break it down step by step.

### Step 1: Simplify \( 2^2 \)
First, we know that \( 2^2 = 4 \).

### Step 2: Substitute into the expression
Now, we can rewrite the expression:
\[ \sqrt[4]{\sqrt[3]{4}} \]

### Step 3: Simplify \( \sqrt[3]{4} \)
Next, we need to simplify \( \sqrt[3]{4} \). We can express 4 as \( 2^2 \):
\[ \sqrt[3]{4} = \sqrt[3]{2^2} \]
Using the property of exponents, this can be rewritten as:
\[ \sqrt[3]{2^2} = 2^{2/3} \]

### Step 4: Substitute back into the expression
Now, we substitute \( \sqrt[3]{4} \) back into the expression:
\[ \sqrt[4]{2^{2/3}} \]

### Step 5: Simplify \( \sqrt[4]{2^{2/3}} \)
Using the property of exponents again, we can rewrite this as:
\[ \sqrt[4]{2^{2/3}} = 2^{(2/3) \cdot (1/4)} = 2^{2/12} = 2^{1/6} \]

### Final Answer
Thus, the final answer is:
\[ 2^{1/6} \]