To simplify the expression \( x^{1/6} \cdot x^{1/6} \), we can use the property of exponents which states that when you multiply two powers with the same base, you add their exponents:
\[ x^{a} \cdot x^{b} = x^{a + b} \]
In this case, we have:
\[ x^{1/6} \cdot x^{1/6} = x^{1/6 + 1/6} = x^{2/6} = x^{1/3} \]
Now, we need to express \( x^{1/3} \) in terms of roots:
\[ x^{1/3} = \sqrt[3]{x} \]
Now, we can match this with the provided options:
- \( \sqrt[3]{x} \)
- \( \frac{1}{\sqrt[3]{x}} \)
- \( \sqrt{x} \)
- \( \sqrt{36} \cdot \sqrt{x} \)
The correct equivalent expression for \( x^{1/3} \) is:
\[ \text{root index 3 Start Root x End Root} \]
So, the answer is \( \sqrt[3]{x} \).