https://file.aiquickdraw.com/mathbot/file/1727198708132t5d1472j.jpg

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!

Запишіть вираз у вигляді степеня з основою а: a^(1/5) * a^(2/5)

Learn how to simplify the expression a^(1/5) * a^(2/5) using the laws of exponents. This step-by-step solution explains the process of adding exponents when multiplying terms with the same base. Suitable for students in Grades 8-10.

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