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Explain me how to get from the red to the blue

The red-circled expression is:

\[
y = x \sqrt{2}
\]

and the blue-circled expression is:

\[
y = \sqrt{\frac{x^2}{2}}
\]

To explain how to transform the red expression into the blue one, let’s break it down step by step:

### 1. Start with the red expression:
\[
y = x \sqrt{2}
\]

### 2. Rewrite \( x \sqrt{2} \) as \( \sqrt{x^2 \cdot 2} \):
Since \( x \sqrt{2} \) can be rewritten as:
\[
x \sqrt{2} = \sqrt{x^2} \cdot \sqrt{2} = \sqrt{x^2 \cdot 2}
\]

### 3. Factor the square root of the product:
\[
\sqrt{x^2 \cdot 2} = \sqrt{\frac{x^2}{2} \cdot 2^2}
\]

### 4. Separate the square roots:
\[
\sqrt{x^2 \cdot 2} = \frac{\sqrt{x^2}}{\sqrt{2}} = \sqrt{\frac{x^2}{2}}
\]

Thus, \( x \sqrt{2} \) simplifies to \( \sqrt{\frac{x^2}{2}} \).

This shows that the red-circled expression \( y = x \sqrt{2} \) is equivalent to the blue-circled expression \( y = \sqrt{\frac{x^2}{2}} \).

Let me know if you would like more details or further clarification!

### Related Questions:
1. How does the property of square roots help in simplifying algebraic expressions?
2. What are the rules for multiplying and dividing square roots?
3. Why does \( \sqrt{x^2} \) equal \( x \) (for positive values of \( x \))?
4. How can this transformation be applied to other similar equations?
5. How would the expression change if we had \( y = x \sqrt{3} \) instead?

### Tip:
Always remember that \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) is a useful identity when simplifying square roots!

This guide explains how to transform the algebraic expression x√2 into √(x^2 / 2), covering key concepts such as radical simplification and the product rule for square roots. Suitable for students in Grades 9-12.

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