Taking the root of a number reverses the operation of raising a base number to a certain power so the answer will be the original base number. Since a number can be multiplied by itself any number of times, there are also any number of different roots. Here though, you are going to concentrate on square roots – the reverse of squaring or raising a base number to the power of two.
If you take the square root of 25, written as , the answer is 5, the original base number. You can check this by reversing the root taken and squaring the result. So, squaring gives 25 – the original number.
However, with any square root of a positive number there is not just one possible answer.
Looking at the original example of the square root of 25, the answer was 5 because . But there is another square root because is also 25. So, any positive number has two square roots, a positive one and a negative one.
To avoid confusion the following convention is used to distinguish between these two outcomes. When both roots are relevant is used to distinguish between these two outcomes (where the symbol is read as ‘plus or minus’). Hence, for both the negative and positive root of 25 you would write:
When only the positive root is relevant is used without the . Hence for the positive square root of 25 you would write:
But how do you actually work out square roots? This usually relies on one of two methods. The first is just memory, and the more you use roots the more you will remember what they are, but this only helps in a few situations. The other method is to use a calculator to do the work for you. If you continue your study of maths, you may well come across methods to work out square roots from scratch as well, but this is definitely something for another time.
To use a calculator to find square roots, use the key.
Try using your calculator to find . Depending on what calculator you are using, you will either have to press the square root button before you enter the number, or after.
Hopefully you should have obtained 25 as the answer.
It is worth noting that a calculator will only ever give you the positive square root, so it can be easy to forget there is another answer as well!
As well as the form of notation covered here, roots can also be shown using power notation. The next section will look at how this is done.