Find the square root of the following numbers using the long division method 21316

To find the square root of **21316** using the **long division method**, follow these steps:

1. **Step 1: Group the digits in pairs**
   Starting from the decimal point and going both left and right (though here we only have a whole number).
   - 21316 → (213)(16)

2. **Step 2: Find the largest number whose square is less than or equal to the first group (213)**
   - \(14^2 = 196\), and \(15^2 = 225\)
   - So, the largest perfect square less than or equal to 213 is 196, and the square root is **14**.

3. **Step 3: Subtract the square and bring down the next group**
   - Subtract 196 from 213: \(213 - 196 = 17\)
   - Bring down the next group (16), so now we have 1716.

4. **Step 4: Double the quotient**
   - Double the quotient (14): \(2 \times 14 = 28\).
   - Now find the largest digit \(x\) such that \((280 + x) \times x\) is less than or equal to 1716.

5. **Step 5: Find \(x\)**
   - Try \(x = 6\):
     \[
     (280 + 6) \times 6 = 286 \times 6 = 1716
     \]
   - This fits exactly.

6. **Step 6: Conclusion**
   - The square root of 21316 is **146**.

Let me know if you need further clarification on the steps or have any questions!

### Related Questions:
1. How can you verify the result of a square root calculation?
2. What is the long division method used for calculating square roots of decimals?
3. How does the long division method compare to using a calculator for square roots?
4. Can you use the long division method for square roots of non-perfect squares?
5. How does the long division method work for larger numbers?

**Tip**: When finding square roots manually, always check intermediate steps for accuracy to avoid calculation errors.

Learn how to find the square root of 21316 using the long division method. This step-by-step guide covers key arithmetic concepts, including grouping digits, finding approximate values, and iterative calculations. Perfect for students in Grades 6-8.

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