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To factor the square root of \( \sqrt{200} \), let's simplify step-by-step:

1. **Factorize 200**:
   \[
   200 = 2^3 \cdot 5^2
   \]

2. **Group terms for square roots**:
   Rewrite \( \sqrt{200} \) as:
   \[
   \sqrt{200} = \sqrt{2^3 \cdot 5^2}
   \]

3. **Separate perfect squares**:
   Break this into:
   \[
   \sqrt{200} = \sqrt{(2^2 \cdot 5^2) \cdot 2}
   \]

4. **Simplify the square root**:
   Extract the square roots of the perfect squares:
   \[
   \sqrt{200} = \sqrt{2^2} \cdot \sqrt{5^2} \cdot \sqrt{2} = 2 \cdot 5 \cdot \sqrt{2}
   \]

5. **Final simplified form**:
   \[
   \sqrt{200} = 10\sqrt{2}
   \]

Would you like me to explain any step in more detail?

### Related Questions:
1. How do you identify perfect square factors in a number?
2. What is the difference between simplifying and evaluating a square root?
3. How would the process differ if the number inside the square root were a prime number?
4. Can you explain how to simplify \( \sqrt{50} \) using a similar method?
5. How do square root simplifications apply to real-world problems?

### Tip:
Always start by identifying the largest perfect square that divides the number to simplify square roots efficiently.

Learn how to factor and simplify √200 using the properties of square roots. This detailed guide explains how to break down the square root of 200 into 10√2 by factoring perfect squares. Suitable for Grades 6-8.

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