`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 simplify it step by step.

### Step 1: Rewrite the expression using exponents
The expression can be rewritten using exponents:
\[ \sqrt[4]{\sqrt[3]{2^{2}}} = \sqrt[4]{(2^{2})^{\frac{1}{3}}} \]

### Step 2: Apply the power of a power rule
Using the power of a power rule, which states that \( (a^m)^n = a^{m \cdot n} \), we can simplify further:
\[ (2^{2})^{\frac{1}{3}} = 2^{2 \cdot \frac{1}{3}} = 2^{\frac{2}{3}} \]
Now, substitute this back into the expression:
\[ \sqrt[4]{2^{\frac{2}{3}}} \]

### Step 3: Apply the power of a power rule again
Now we apply the power of a power rule again:
\[ \sqrt[4]{2^{\frac{2}{3}}} = (2^{\frac{2}{3}})^{\frac{1}{4}} = 2^{\frac{2}{3} \cdot \frac{1}{4}} = 2^{\frac{2}{12}} = 2^{\frac{1}{6}} \]

### Final Answer
Thus, the expression \( \sqrt[4]{\sqrt[3]{2^{2}}} \) simplifies to:
\[ 2^{\frac{1}{6}} \]