The LessonThe 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 DefinitionThe 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 RootThe square root of 36 is written √36. The square root of 36 (√36), when multiplied by itself, gives 36:
√36 × √36 = 36The 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 NumberFinding 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 NumberIt 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 NumbersA 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").
- 1 = 12 ∴ √1 = 1.
- 4 = 22 ∴ √4 = 2.
- 9 = 32 ∴ √9 = 3.
Square Roots If It Is Not a Square NumberIf 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 AlgebraIn 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)
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 RootsThere 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 DecimalIt 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 = 4What is √5? 5 is between 4 and 9. We can therefore say √5 must be between √4 and √9, or between 2 and 3.