To calculate \(\sqrt{424}\), you can follow these steps:

1. **Estimate the Range:**
   - Find two perfect squares between which 424 lies.
   - \(20^2 = 400\) and \(21^2 = 441\), so \(\sqrt{424}\) is between 20 and 21.

2. **Use Approximation:**
   - You can start with a rough estimate, say, 20.5, and square it:
     \[
     20.5^2 = 420.25
     \]
   - Since \(420.25\) is close to \(424\), but less, the square root must be slightly more than 20.5.

3. **Refine the Approximation:**
   - Try a value like 20.6:
     \[
     20.6^2 = 424.36
     \]
   - This is very close to 424. Hence, \(\sqrt{424} \approx 20.6\).

4. **Exact Calculation:**
   - For more precision, use a calculator to find:
     \[
     \sqrt{424} \approx 20.59126
     \]
   - This value can be rounded depending on the required precision.

Would you like more details or have any questions? Here are some related questions you might find interesting:

1. How do you calculate square roots by hand using the long division method?
2. What are some common methods for approximating square roots without a calculator?
3. How can you estimate square roots using linear interpolation?
4. Why is the square root of a non-perfect square always an irrational number?
5. What are the applications of square roots in real-life problems?

**Tip:** When approximating square roots, always check how close your squared value is to the original number to refine your estimate further.

Learn how to calculate the square root of 424 with this detailed step-by-step guide. This tutorial covers estimation, approximation, and the use of calculators for precision, suitable for students in Grades 8-10.

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