Root
The root of a number x is another number, which when multiplied by itself a given number of times, equals x.

For example, the third root (also called the cube root) of 64 is 4, because if you multiply three fours together you get 64:

4 × 4 × 4 = 64
This would be written as The above would be spoken as "the third root of 64 is 4" or "the cube root of 64 is 4".

• The second root is usually called the "square root".
• The third root of a number is usually called the "cube root",
• After that, they are called the nth root, for example the 5th root, 7th root etc

## Sometimes there are two roots

For every even-degree root (for example the 2nd, 4th, 6th ....) there are two roots. This is because multiplying two positive or two negative numbers both produce a positive result. For example, consider the square root of 9.

What number, multiplied by itself will produce 9?
Obviously 3 will work:

3 × 3 = 9
But so will -3:
-3 × -3 = 9

So to be precise, we say the "square root of 9 is plus and minus 3". This is written: The symbol in front of the 3 is read as "plus or minus".

This is true for every even order root, for example the 2nd, 4th, 6th and so on.

## Sometimes there are no roots

When the root is of even degree ( for example the 2nd, 4th, 6th ....) you cannot find the root of a negative number. This is because there is no number, multiplied by itself, that produces a negative result*. For example, if you tried to find the square root of -9, you cannot do it. Both +3 and -3 would produce +9 when multiplied together.

With odd-degree roots such as the 3rd, fifth etc, you can find the root of a negative number and the root will have the same sign as the radicand. For example the cube root of -27 is -3, because

-3 × -3 × -3 = -27

## The symbols

 Degree The number of times the radicand is multiplied by itself. 2 means square root, 3 means cube root. After that they are called the 4th root, 5th root and so on. If this is missing, it is assumed to be 2 - the square root. Radical symbol The √ symbol that means "root of". The length of the horizontal bar is important. See note below. Radicand The thing you are finding the root of.

## Another way to write it

Roots can also be written in exponent form. In general So for example the cube root of x would be written Which would be pronounced "x to the power of one third".

* There is a way around this by using what are called "complex numbers".
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