The error your classmate made is that they estimated 397 itself to be around 200, instead of estimating the square root of 397.

Here's a clearer breakdown:

1. The square root of 397 (i.e., $\sqrt{397}$) is approximately 19.93, not anywhere near 200.
2. Your classmate likely confused finding the square root with estimating the value of 397 itself, hence why they arrived at a value near 200.
3. To correct this mistake, the classmate should have recognized that the square root of a number is much smaller than the number itself. Since $\sqrt{400} = 20$, it gives a good estimation that $\sqrt{397}$ is very close to 20.

Would you like more details or explanations on this?

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Here are 5 related questions to expand on this topic:

1. What is the difference between squaring a number and taking its square root?
2. How can we estimate square roots of non-perfect squares like 397?
3. What are the square roots of other numbers near 397, such as 400 or 390?
4. Why do square roots of larger numbers get closer to integers?
5. How does the error in estimating square roots affect real-life applications?

Tip: When estimating square roots, find nearby perfect squares (like 400) to get a rough idea of the value.