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# How to square root any number without a calculator using a trial and improvement method.

Here I will show you a method on how to square root a number which isn’t a square number without using a calculator. Firstly let me remind you what a square root is - basically, you are looking for a number when multiplied by itself, which gives you the number that you are square rooting. For example, the square root of 81 is 9 because 9 x 9 = 81.

So all you need to do is solve the equation x^{2} = “something”. You can do this by using a trial and improvement method. You will have to be good at long multiplication to carry out this method as you can’t use your calculator!

Let’s take a look at a couple of examples.

**Example 1 **

Work out the square root of 40 without a calculator. Give your answer to 1 decimal place.

First you will need to figure out which two square numbers the number you are square rooting is between.

Since 6 squared is 36 and 7 squared is 49 the square root of 40 is between 6 and 7.

Next go halfway between 6 and 7 which is 6.5 and work out 6.5 squared:

6.5^{2} = 6.5 x 6.5 = 42.25

6.5 is too big so let’s try 6.4 squared:

6.4^{2} = 6.4 x 6.4 = 40.96

Like before, 6.4 is too big, so let’s try 6.3 squared:

6.3^{2} = 6.3 x 6.3 = 39.69

So 6.3 is too small and 6.4 is too big. However, since 6.3 squared is closer to 40 then the square root of 40 is 6.3 to 1 decimal place.

**Example 2 **

Work out the square root of 96 without a calculator. Give your answer to 1 decimal place.

Just like the first example, you need to figure out which two square numbers the number you are square rooting is between.

Since 9 squared is 81 and 10 squared is 100 the square root of 96 is between 9 and 10.

Next go half way between 9 and 10 which is 9.5 and if you square 9.5 you get:

9.5^{2} = 9.5 x 9.5 = 90.25

9.5 is too small so let’s try 9.6 squared:

9.6^{2} = 9.6 x 9.6 = 92.16

9.6 is too small again so let’s try 9.7 squared:

9.7^{2} = 9.7 x 9.7 = 94.09

9.7 is too small again so let’s try 9.8 squared:

9.8^{2} = 9.8 x 9.8 = 96.04

So 9.7 is too small and 9.8 is too big. However, since 9.8 squared is closer to 96 then the square root of 96 is 9.8 to 1 decimal place.

As you can see finding the square root of number without a calculator can be done reasonably quickly. However, if you are asked for a greater degree of accuracy then this can be much more time consuming.