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can you explain to me how to get from the first example to the second one

To help explain how to get from the first expression to the second one, I'll describe both and then break down the transformation.

1. **First expression:**
   \[
x \times \sqrt{2}
   \]
   This is the product of \(x\) and the square root of 2.

2. **Second expression:**
   \[
\sqrt{\frac{x^2}{2}}
   \]
   This is the square root of the fraction \(\frac{x^2}{2}\).

### Transformation Explanation:

1. Start with the first expression: 
   \[
   x \times \sqrt{2}
   \]

2. Square the expression inside the square root:
   \[
   (x \times \sqrt{2})^2 = x^2 \times 2
   \]
   When you square both terms inside, you get \(x^2 \times 2\).

3. Divide by 2 to adjust for the square:
   \[
   \frac{x^2}{2}
   \]
   This fraction is now simplified.

4. Take the square root of the entire fraction:
   \[
   \sqrt{\frac{x^2}{2}}
   \]
   This matches the form of the second expression.

### Summary:
The key step is rewriting the product \(x \times \sqrt{2}\) as a fraction under a square root, squaring the whole expression, and then simplifying to arrive at the form in the second expression.

Would you like to ask any more details or questions?

### Related Questions:
1. How does squaring affect the product inside the square root?
2. Can this method be applied to other products with square roots?
3. What are the rules for simplifying square roots of fractions?
4. How do you handle negative values of \(x\) in these expressions?
5. What happens if the number inside the square root is not 2?

### Tip:
Always remember that squaring a square root eliminates the square root, making simplifications easier.

Learn how to simplify the expression x × √2 into √(x^2 / 2) with step-by-step explanations. This guide covers the algebraic manipulation of square roots and is suitable for high school students, Grades 9-12.

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