Let's simplify the given expression step by step:

$\sqrt{2\sqrt{4^x}}$

1. Start with the inner expression $\sqrt{4^x}$: $\sqrt{4^x} = (4^x)^{1/2} = 4^{x/2}$

2. Substitute this back into the original expression: $\sqrt{2\sqrt{4^x}} = \sqrt{2 \cdot 4^{x/2}}$

3. Write $4^{x/2}$ as $(2^2)^{x/2}$, which simplifies using exponent rules: $(2^2)^{x/2} = 2^{2 \cdot (x/2)} = 2^x$

   So, $\sqrt{2 \cdot 4^{x/2}} = \sqrt{2 \cdot 2^x}$.

4. Combine $2$ and $2^x$ using the property of exponents: $2 \cdot 2^x = 2^{1 + x}$

5. Now the expression becomes: $\sqrt{2^{1 + x}}$

6. Apply the square root to the exponent: $\sqrt{2^{1 + x}} = 2^{(1 + x)/2}$

Thus, the simplified expression is:

$2^{(1 + x)/2}$

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Do you have any questions about this solution or want to explore related concepts? Here are five questions you might find helpful to expand your understanding:

1. How do exponentiation and roots interact when simplifying expressions?
2. What are some other examples of simplifying nested radicals?
3. How can we verify the correctness of such simplifications using specific values of $x$?
4. What are the properties of exponential expressions that make them useful in solving equations?
5. How does the base of the exponent affect the simplification process?

Tip: Always break down nested expressions step by step, starting from the innermost part, to avoid errors in simplification.