Question

What are the different methods of finding the square root of perfect squares?

Answer 1

There are a number of ways to calculate square roots without a calculator. Here is guess and divide method.
Step–1: Estimate/find a perfect square root as close as possible to your number.
Step–2: Divide your number by the square root.
Step–3: Calculate the average of the result of step 2 and the root.
Step–4: Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.
Calculate the square root of 10 (√ 10) to 4 decimal places.
1.    Find the perfect square number closer to 10. 32 = 9 and 42 = 16, so take 3.
2.    Divide 10 by 3. 10÷3 = 3.33 (you can round off the answer)
3.    Average 3.33 and 3. (3.33 + 3)÷2 = 3.1667
Repeat step 2: 10÷3.1667 = 3.1579 
Repeat step 3: Average 3.1579 and 3.1667.(3.1579 + 3.1667)÷2 = 3.1623
Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001
If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.
Here's another example, √695 =? Let us first guess for the root as 25.695 ÷ 25 = 27.8(25 + 27.8) ÷ 2 = 26.4695 ÷ 26.4 = 26.3257(26.4 + 26.3257) ÷ 2 = 26.36285695 ÷ 26.36285 = 26.3628553
Last two findings are about equal, so you now have a good estimate of the root of 695. You can keep going through the process until you are satisfied with the accuracy.
So, √24.6=?
Let's try 5 since 52 = 25, which is pretty close to 24.6.Divide 24.6 by 5. 24.6 ÷ 5 = 4.92(5 + 4.92) ÷ 2 = 4.96 You can stop here. 4.96 is pretty close to 4.9598 which is the actual square root of 24.6. Repeat steps 2 and 3 to any desired level of accuracy. The further you go, the harder the long division becomes. But the first few cycles yield a pretty close answer.24.6 / 4.96 = 4.9596(4.96 + 4.9596) ÷ 2 = 4.9598