The expression you're asking about is:

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

Let's break this down step by step:

1. $\sqrt{4^x}$ can be rewritten using the property of square roots: $\sqrt{a^b} = a^{b/2}$. So, $\sqrt{4^x} = 4^{x/2}$.
2. Now, you have the expression: $\sqrt{2} \times 4^{x/2}$.

$\sqrt{2}$ is simply $2^{1/2}$, so we can now express the whole thing as:

$2^{1/2} \times 4^{x/2}$

Since $4 = 2^2$, we substitute that into the expression:

$2^{1/2} \times (2^2)^{x/2}$

Applying the exponent rule $(a^m)^n = a^{m \cdot n}$, this becomes:

$2^{1/2} \times 2^{x}$

Now, using the property of exponents $a^m \times a^n = a^{m+n}$, you can add the exponents:

$2^{1/2 + x}$

Thus, the simplified expression is:

$2^{x + 1/2}$

This is the final simplified form of the given expression.

Would you like more details or have any questions? Here are five related questions:

1. What is the square root of a product of numbers?
2. How does exponentiation distribute over multiplication?
3. How would this expression change if the base was different (e.g., cube roots)?
4. What happens if $x$ is a specific value (e.g., $x = 2$)?
5. Can you graph $2^{x + 1/2}$?

Tip: When multiplying square roots, remember they can often be rewritten in exponential form, which simplifies combining terms.