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

Let's break down the steps to understand how the given expression transforms from the first equation to the second.

### Given:
\[
\sqrt{x^2 \cdot 2} = \sqrt{\frac{x^2}{2} \cdot 2^2}
\]

### Step-by-Step Explanation:

1. **Original Expression**: 
   \[
   \sqrt{x^2 \cdot 2}
   \]
   This is the starting point, where the square root is taken over the product of \(x^2\) and 2.

2. **Factor and Rewrite the Expression**:
   We need to manipulate the terms inside the square root so that it matches the form on the right-hand side. Notice that on the right-hand side, there is a fraction involving \(x^2\) and 2. 

   Start by considering how to express \(2\) in terms of a product involving fractions:
   \[
   x^2 \cdot 2 = x^2 \cdot \left(\frac{2}{2}\right) \cdot 2
   \]
   Now, separate the factors as:
   \[
   = \left(\frac{x^2}{2}\right) \cdot (2 \cdot 2)
   \]
   This simplifies to:
   \[
   = \frac{x^2}{2} \cdot 2^2
   \]

3. **Taking the Square Root**:
   Now that we have rewritten the expression under the square root:
   \[
   \sqrt{x^2 \cdot 2} = \sqrt{\frac{x^2}{2} \cdot 2^2}
   \]

This matches the right-hand side of the equation provided.

### Summary:
The key transformation involves expressing the number 2 inside the square root in a way that introduces a fraction, which allows us to match the desired form by manipulating the factors.

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Would you like more details or have any other questions?

### Related Questions:
1. What is the general rule for distributing square roots over products?
2. How do you handle square roots of fractions in expressions?
3. Can you explain how to factor and simplify square roots with algebraic terms?
4. What happens when you take the square root of a squared term?
5. How can we apply this method to more complex expressions?

### Tip:
When simplifying expressions involving square roots, always look for opportunities to factor and reorganize terms under the square root sign to match the desired form.

Can you explain to me how to get from the first one to the second one?

Learn how to simplify the square root of x^2 times a constant with a detailed step-by-step solution. This example covers key algebraic concepts, including square root properties and factoring techniques. Suitable for students in Grades 9-12.

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