How to do square roots in head?

You can estimate square roots in your head using an iterative approximation method. Here's how:

Steps to Estimate Square Roots Mentally

This method relies on making an initial guess and then refining it.

1. Choose a nearby guess: Find a perfect square close to the number you want to find the square root of. For example, to find the square root of 59.6, you know that 8 * 8 = 64, which is close. So, you can start with a guess of approximately 8. \
   For example: √59.6 ≈ 8

2. Calculate the difference: Determine the difference between the square of your guess and the original number. \
   For example: 8² – 59.6 = 64 - 59.6 = 4.4

3. Divide by twice the guess: Divide the difference you calculated in the previous step by twice your initial guess. This gives you a correction factor. \
   For example: 4.4 / (2 * 8) = 4.4 / 16 = 0.275

4. Apply the correction: Add or subtract the correction factor from your initial guess to get a better estimate. If your guess squared was larger than the original number, you subtract the correction. If it was smaller, you add the correction. \
   For example: Since 8² was greater than 59.6, subtract the correction: 8 – 0.275 = 7.725

Therefore, √59.6 ≈ 7.725

Example Summary in Table Format

Step	Calculation	Result
Initial Guess	√59.6 ≈ guess	8
Difference Calculation	82 – 59.6	4.4
Divide by Twice Guess	4.4 / (2 * 8)	0.275
Apply Correction	8 – 0.275	7.725