Square Root Numbers and Other Roots
For numbers, a root is another number that produces that whole number when raised to a certain
power/exponent.
With the number 8 for example:
2 is the 3rd root of
8, =>
2^{3} =
2 ×
2 × 2 =
8
The 3rd root is commonly referred to as the cube root.
However, the most common root that most will encounter is likely to be the square root of
numbers.
The square root is the 2nd root of a number.
So it is a number that multiplies with itself once.
4 is the ;2nd root of
16, =>
4^{2} =
4
×
4 =
16
5 is the 2nd root of
25, =>
5^{2} =
5 ×
5 =
25
Square Root of Numbers and other Roots
Notation
The symbol for the square root of numbers is the radical symbol √.
Numbers featuring this symbol are referred to as radical terms, but can sometimes just be called
radicals for short.
As seen earlier in the page, it's the case that:
√16 =
4 ,
√25
=
5
But negative numbers can also be square roots too:
4 × 4 =
16
So it's actually more accurate to say that 4 is a square root of
16, rather than the square root of 16, as 4 is also a root too.
√16 =
+4
Cube Root & Other Roots
The notation for the cube root is similar to the square root, with the cube root being the 3rd
root, it is denoted by ^{3}√.
The 4th root is denoted by ^{4}√, the 5th root by ^{5}√, and so on.
Calculators today usually have the [ ^{x}√ ] button that can assist in finding specific roots of numbers quickly.
To establish ^{4}√256 .
Type 4, followed by the [ ^{x}√ ] button, then type 256, which gives the answer 4.
[ 4 × 4 × 4 × 4 = 256 ]
Examples
(1.1)
^{3}√512
=
8
(1.2)
^{5}√243 =
3
Even and Odd Roots, General Case
We can generalize the case of root of a number with the same exponent as the root, depending on
whether the base number is positive or negative.
It can be helpful to know these facts, for more advanced situations when required to simplify a
radical containing variables.
Firstly some examples of the square root of a squared number.
√2^{2} =
√4
=
2
√5^{2} =
√25
=
5
Looking at these two examples, it could be reasonable to assume that it's always the case that:
√a^{2} =
a
But this is not quite the case. For example with 3.

3 × 
3 =
9
Thus:
√3^{2}
=
√9 =
3
So to ensure the answer is always positive for an even number root of the same even exponent, the
absolute value of the base number has to be taken.
√a^{2} =
a
,
^{4}√a^{4} =
a etc.
So the general case is
^{n}√a^{n} =
a when "
n" is even.
With the same odd exponent and root however, things are simpler and we don't have to take the exact
value of the base number.

2^{3} = 
2 × 
2 × 
2 = 
8
Thus:
^{3}√2^{3} = 
2
In fact the general case is
^{n}√a^{n} =
a when "
n" is odd.

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