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 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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