To simplify the expression \( x^{\sqrt{\frac{1}{6}}} \cdot x^{\frac{1}{6}} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
- The first term is \( x^{\sqrt{\frac{1}{6}}} \).
- The second term is \( x^{\frac{1}{6}} \).
Now sum the exponents:
\[ \sqrt{\frac{1}{6}} + \frac{1}{6} \]
To add these two fractions, it's helpful to express \(\sqrt{\frac{1}{6}}\) in terms of a fraction. We know that:
\[ \sqrt{\frac{1}{6}} = \frac{1}{\sqrt{6}} \]
Next, rationalize the denominator:
\[ \frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6} \]
Now we add \(\frac{\sqrt{6}}{6}\) and \(\frac{1}{6}\):
\[ \frac{\sqrt{6} + 1}{6} \]
This means:
\[ x^{\sqrt{\frac{1}{6}}} \cdot x^{\frac{1}{6}} = x^{\frac{\sqrt{6} + 1}{6}} \]
None of the provided responses seem to represent this expression directly. Could you please clarify what format or simplification is expected for the answer or recheck the response options?