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What Is a Square Root? (Mathematics Glossary)

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 Merriam-Webster 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 12 ("1 squared").
  • 4 is a square number. It is 2 × 2, or 22 ("2 squared").
  • 9 is a square number. It is 3 × 3, or 32 ("3 squared").
This means that if we square root a square number, we get a whole number:
  • 1 = 12 ∴ √1 = 1.
  • 4 = 22 ∴ √4 = 2.
  • 9 = 32 ∴ √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

√x2 = x

√x × √y = √(xy)
Interactive Test
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Here's a second test on square roots.
Here's a third test on square roots.
Note

Why "Square" Roots?

A square root is the inverse (opposite) of a square number.

If a number represented the area of a square, then the square root represents the length of the side of that square.

A square with an area of 1 has sides of length 1.



A square with an area of 4 has sides of length 2.



A square with an area of 9 has sides of length 3.

There Are Always Two Square Roots

There are always two square roots for each number, a positive root and a negative root.

The square roots of 36 are 6 and -6.

It is conventional just to consider the positive square root, so we would say √36 is 6.

'Guess-timating' Square Roots As a Decimal

It is very difficult to work out a square root as a decimal without a calculator. But, it is possible to have a good guess from the ones we do know.

The ones we know are the square roots of the square numbers:

1² = 1    so     √1 = 1
2² = 4    so     √4 = 2
3² = 9    so     √9 = 3
4² = 16   so    √16 = 4

What is √5?

5 is between 4 and 9.

We can therefore say √5 must be between √4 and √9, or between 2 and 3.