To solve the expression \( x^{1/6} \times x^{1/6} \), we can use the property of exponents that states when multiplying like bases, we add the exponents:
\[ x^{1/6} \times x^{1/6} = x^{(1/6 + 1/6)} = x^{2/6} = x^{1/3} \]
Now, we want to express \( x^{1/3} \) in terms of square roots. We can rewrite \( x^{1/3} \) as:
\[ x^{1/3} = \sqrt[3]{x} \]
At this point, let's check the options given:
- \( \sqrt{x} \)
- \( 36 \sqrt{x} \)
- \( \frac{1}{3} \sqrt{x} \)
- \( 3 \sqrt{x} \)
None of these options directly equate to \( \sqrt[3]{x} \). However, it's important to note that if you meant to provide a relation involving a radical, we could express \( x^{1/3} \) in terms of square roots directly:
\[ \sqrt[3]{x} \text{ as } x^{1/3} \text{ has no direct equivalent among } \sqrt{x}, 36\sqrt{x}, \frac{1}{3}\sqrt{x}, \text{ or } 3\sqrt{x}. \]
Thus, the expression \( x^{1/3} \) produced from multiplying \( x^{1/6} \) with itself does not match any given option correctly.
Please verify if the options are indeed accurate or if there was a typographical error.