Square Roots
What Is Square Root?
The square root of a number, when multiplied by itself, gives that number.
A square root is denoted by writing a √ (radical) symbol in front of a number.
Dictionary Definition
The MerriamWebster dictionary defines a square root as "a factor of a number that when squared gives the number."
A Real Example of a Square Root
The square root of 36 is written √36.
The square root of 36 (√36), when multiplied by itself, gives 36:
√36 × √36 = 36
The square root of √36 is equal to 6. The square root of 36 (6), when multiplied by itself, gives 36:
6 × 6 = 36
The Square Root Is the Opposite of Squaring a Number
Finding a square root is the inverse (opposite) of squaring a number. (Don't forget: Squaring a number means multiplying the number by itself).
Taking the square root of 36 gives √36. Squaring √36 gives 36.
Remember that √36 is 6:
How to Find the Square Root of a Number
It can be difficult to find the square root of a number by hand. (It is easy on a calculator. Just press the √ button!)
A different approach is suggested depending whether you are finding the square root of a square number or not.
Square Roots of Square Numbers
A square number a whole number (an integer) that results from a smaller whole number being multiplied by itself:

1 is a square number. It is 1 × 1, or 1^{2} ("1 squared").

4 is a square number. It is 2 × 2, or 2^{2} ("2 squared").

9 is a square number. It is 3 × 3, or 3^{2} ("3 squared").
This means that if we square root a square number, we get a whole number:

1 = 1^{2} ∴ √1 = 1.

4 = 2^{2} ∴ √4 = 2.

9 = 3^{2} ∴ √9 = 3.
The square roots of the first 10 square numbers are shown below:
Square Roots If It Is Not a Square Number
If number is not a square number, its square root will not be a whole number.
For example, √2 = 1.414213562...
Because √2 cannot be simplified to a whole number (or even a fraction), it is called a surd.
It can be neater and more precise just to leave surds as they are: just write √2 as √2, not 1.414213562...
A surd is an irrational number (it cannot be expressed as a fraction). As a decimal, it goes on forever.
Surds have their own rules, for example:
√2 × √2 = 2
√2 × √3 = √(2 × 3) = √6
Square Roots in Algebra
In algebra, letters are used instead of numbers.
Just as we can find the square root of a number, such as √2...
...we can find the square root of a letter, such as √x.
We would call √x the square root of x.
Square roots of letters have their own rules, for example:
√x × √x = x
√x^{2} = x
√x × √y = √(xy)