The problem in the image asks to simplify the expression involving powers of $a$:

$a^{\frac{1}{5}} \cdot a^{\frac{2}{5}}$

Solution:

To simplify this expression, we apply the rule of exponents that states:

$a^m \cdot a^n = a^{m+n}$

In this case, $m = \frac{1}{5}$ and $n = \frac{2}{5}$. So we add the exponents:

$a^{\frac{1}{5}} \cdot a^{\frac{2}{5}} = a^{\frac{1}{5} + \frac{2}{5}} = a^{\frac{3}{5}}$

Thus, the simplified expression is:

$a^{\frac{3}{5}}$

Final Answer:

The expression in terms of a single power of $a$ is $a^{\frac{3}{5}}$.

Let me know if you would like more details on this! Here are 5 related questions to explore:

1. What is the general rule for multiplying powers with the same base?
2. How would you simplify $a^{\frac{3}{5}} \cdot a^{\frac{4}{5}}$?
3. How do you handle division of exponents with the same base?
4. What happens if the exponents in multiplication have different bases?
5. How would you represent $a^{\frac{3}{5}}$ as a radical expression?

Tip: When multiplying expressions with the same base, always add the exponents to simplify quickly!