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The problem asks to find which expression is equivalent to \( 2^{\frac{1}{4}} \sqrt[4]{2x} \).

Let's break this down step by step:

### Given:
The expression is \( 2^{\frac{1}{4}} \sqrt[4]{2x} \).

The radical \( \sqrt[4]{2x} \) can be rewritten using rational exponents. Recall that:
\[
\sqrt[4]{2x} = (2x)^{\frac{1}{4}}.
\]
Thus, the entire expression becomes:
\[
2^{\frac{1}{4}} \times (2x)^{\frac{1}{4}}.
\]

### Simplifying the product:
We can combine the terms:
\[
2^{\frac{1}{4}} \times (2x)^{\frac{1}{4}} = (2 \times 2x)^{\frac{1}{4}} = (4x)^{\frac{1}{4}}.
\]

Therefore, the equivalent expression is \( (4x)^{\frac{1}{4}} \).

### Correct answer:
The correct choice is the second option, \( (4x)^{\frac{1}{4}} \).

Would you like any further explanation or help with related concepts? Here are 5 questions to deepen your understanding:

1. How do you rewrite radicals as expressions with fractional exponents?
2. Can you explain the rules for multiplying exponents with the same base?
3. How would you simplify \( (ab)^n \) for different values of \( n \)?
4. What is the difference between a square root and a fourth root in terms of exponents?
5. How would you approach solving equations that involve fractional exponents?

### Tip:
When simplifying expressions involving exponents and roots, remember that roots can always be converted to fractional exponents for easier manipulation.

Which expression is equivalent to 2^(1/4) * 4√(2x)?

Learn how to simplify rational exponents and radicals in algebra. This problem demonstrates converting roots to fractional exponents and simplifying expressions step-by-step, making it perfect for high school students (Grades 10-12).

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