Root (of a number)
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
When there are two roots like this, unless stated otherwise we mean the positive one. So strictly speaking, when we write
√4, we mean the positive root, +2. This is called the 'principal root'.
Roots of negative numbers
There are no real even-order roots of negative numbers. For example there is no real square root of -9, because
-3 × -3 =+9, and +3 × +3 =+9 also. This applies to all even-order roots, 2nd (square) root, 4th root, 6th root and so on.
However, there are odd-order roots of negative numbers. For example –3 is a cube root of –27.
This is because
–3 × –3 × –3 = –27.
The first two terms when multiplied produce +9, then the next multiply is
+9 × –3 = –27.
This applies to all odd-order roots such as 3rd (cube) root, 5th root 7th root etc.
It states above that there is no real square root of a negative number. Note the word 'real'. What this is saying is that there is no
that is the square root of a negative number.
However, in math and engineering we frequently have the need to find the square root of a negative number. To solve this, we introduce the idea of the 'imaginary' number. It involves the symbol i which stands for the square root of negative one.
Or put another way, i2 = –1
In use , we can use it to express the square root of any negative number. For example
This means that the square root of –25 is the square root of +25 times the square root of negative one.
For more on imaginary number see
||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.
||The √ symbol that means "root of". The length of the horizontal bar is important. See note below.
||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".
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Other exponents and roots topics
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