To simplify the expression \( 9^{1/5} \cdot 9^{1/2} \), you can use the property of exponents that states:
\[ a^m \cdot a^n = a^{m+n} \]
In this case, \( a = 9 \), \( m = \frac{1}{5} \), and \( n = \frac{1}{2} \). Therefore, we can add the exponents:
\[ 9^{1/5} \cdot 9^{1/2} = 9^{1/5 + 1/2} \]
Now, we need to add \( \frac{1}{5} + \frac{1}{2} \). First, find a common denominator. The least common multiple of 5 and 2 is 10.
Convert each fraction:
\[ \frac{1}{5} = \frac{2}{10} \quad \text{and} \quad \frac{1}{2} = \frac{5}{10} \]
Now add the fractions:
\[ \frac{2}{10} + \frac{5}{10} = \frac{7}{10} \]
So, we have:
\[ 9^{1/5 + 1/2} = 9^{7/10} \]
Thus, the simplified expression is:
\[ 9^{7/10} \]