Square Root, Step by Step

A calculator gives you the answer, but it does not show you how. This tool works a square root out the long way and reveals each step, so you can follow the method - or play it through to see it unfold. Pick the way that suits you:

• Regular - the long-division method taught in school, where you find the root one digit at a time. It is exact and works for any whole number.
• Fast approximation - a guess-and-improve method (start with a rough guess, then average it with the number divided by the guess, again and again). It closes in on the answer in a few steps and handles decimals too.

How to Use It

1. Enter a number.
2. Choose Regular or Fast approximation. For the regular method you can also set how many decimal places to work out.
3. Use Next step to go one step at a time, Auto-play to watch it run, or Show all to reveal the whole working at once.

The Two Methods

Regular (long division). Group the digits in pairs from the decimal point. For each pair you find the largest digit that fits, double the part of the root found so far to build the next trial number, subtract, and bring down the next pair. The root grows one digit at a time, and you can keep going past the decimal point for as many places as you want. Because each digit is the largest that fits, the result is shown truncated to the places you ask for, not rounded - so the last digit can be one lower than a calculator that rounds (for example √2 to six places shows 1.414213).

Fast approximation (guess and average). Start with any guess. A better guess is the average of your guess and the number divided by your guess. Repeat, and the estimate doubles its accuracy roughly every step - so even a rough start is correct to many places within a handful of rounds.

Related Tools

• Just want the answer? Use the square root calculator.
• Find cube roots with the cube root calculator.
• Practice powers and roots with the exponents worksheet generator.