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.