Math Problem Statement

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

Solution

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

a15a25a^{\frac{1}{5}} \cdot a^{\frac{2}{5}}

Solution:

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

aman=am+na^m \cdot a^n = a^{m+n}

In this case, m=15m = \frac{1}{5} and n=25n = \frac{2}{5}. So we add the exponents:

a15a25=a15+25=a35a^{\frac{1}{5}} \cdot a^{\frac{2}{5}} = a^{\frac{1}{5} + \frac{2}{5}} = a^{\frac{3}{5}}

Thus, the simplified expression is:

a35a^{\frac{3}{5}}

Final Answer:

The expression in terms of a single power of aa is a35a^{\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 a35a45a^{\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 a35a^{\frac{3}{5}} as a radical expression?

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

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Math Problem Analysis

Mathematical Concepts

Exponents
Laws of Exponents
Algebra

Formulas

a^m * a^n = a^(m+n)

Theorems

Exponentiation rules

Suitable Grade Level

Grades 8-10