To simplify the expression \( 5^{1/3} \cdot 5^{1/6} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
Here, we have:
\[ 5^{1/3} \cdot 5^{1/6} = 5^{(1/3 + 1/6)} \]
Next, we need to add the exponents \( \frac{1}{3} \) and \( \frac{1}{6} \). To do this, we'll find a common denominator. The least common multiple of 3 and 6 is 6.
Convert \( \frac{1}{3} \) to have a denominator of 6:
\[ \frac{1}{3} = \frac{2}{6} \]
Now, add \( \frac{2}{6} \) and \( \frac{1}{6} \):
\[ \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]
Now we can simplify the expression:
\[ 5^{1/3} \cdot 5^{1/6} = 5^{1/2} \]
Thus, the simplified expression is:
\[ 5^{1/2} = \sqrt{5} \]
So, the final answer is:
\[ \sqrt{5} \]