To simplify the expression \( 2(2x^{1/5})^{1/5} \), follow these steps:

1. **Simplify the inner expression**: Start by simplifying the expression inside the parentheses.

   \[
   (2x^{1/5})^{1/5} 
   \]

   When you raise a power to another power, you multiply the exponents. So:

   \[
   (2x^{1/5})^{1/5} = 2^{1/5} \cdot (x^{1/5})^{1/5} = 2^{1/5} \cdot x^{1/25}
   \]

2. **Distribute the factor of 2**: Now, multiply the simplified expression by 2.

   \[
   2 \cdot (2^{1/5} \cdot x^{1/25}) = 2^{1+1/5} \cdot x^{1/25}
   \]

   Simplify the exponent of 2:

   \[
   2^{1+1/5} = 2^{5/5 + 1/5} = 2^{6/5}
   \]

3. **Final simplified expression**:

   \[
   \boxed{2^{6/5} \cdot x^{1/25}}
   \]

So, the simplified form of the expression \( 2(2x^{1/5})^{1/5} \) is \( 2^{6/5} \cdot x^{1/25} \).

Would you like further details, or do you have any questions?

Here are 5 related questions:
1. How do you simplify an expression with fractional exponents?
2. What is the general rule for multiplying powers with the same base?
3. How can you simplify a power raised to another power?
4. What happens when you multiply two exponents with the same base?
5. How do you express a radical in exponential form?

**Tip:** Remember that fractional exponents can be interpreted as roots. For example, \( x^{1/n} \) is equivalent to the \( n \)-th root of \( x \).

Learn how to simplify the expression 2(2x^(1/5))^(1/5) with detailed steps and explanations. This problem involves exponentiation and fractional exponents, suitable for advanced high school students. Get clarity on power rules and simplify mathematical expressions effectively.

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