Square root of 2 multiplied by square root of 4 to the x power

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.

Explore how to simplify the expression √2 * √4^x using exponent rules and radical simplification techniques. This step-by-step guide covers key concepts in simplifying square roots and exponents, suitable for Grades 9-11.

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