square root formula


In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square (the result of multiplying the number by itself) is x. Every non-negative real number y has a unique non-negative square root, called the principal square root, denoted by a radical sign as . For positive x, the principal square root can also be written in exponent notation, as x1/2.  

Calculating square roots:

If n = m2, then we call m a square root of n. Hence, a square root of 4 is 2, since 4 = 22.

Therefore formula for square root of a number m is given by:


Similarly, the square roots value of ?25 are 5 and –5, since 25 = 52 and 25 = (–5)2 . Similarly, square root of 49 is 7; square root of 121 is 11, etc. hence, if n is a perfect square, then its square root is an integer. If n is not a perfect square, then there is no integer m such that square root of n is m, i.e., it does not have an integral square root. Square stands for perfect square and square root are known as integral square root. Based on the properties of square numbers we have the following propositions about square roots:

  If the unit’s digit of a number is 2, 3, 7 or 8 after that it is not a perfect square and hence does not have a square root.

  If a number has a square root, then its unit’s digit must be 0, 1, 4, 5, 6 or 9.

  If a number ends in an odd number of zeros, then it does not have a square root. If a square number is followed by an even number of zeros, it has a square root. In addition, the number of zeros at the end of the square root is half the number of zeros in the number.

  The square root of an even square number is even and the square root of an odd square number is odd.


Therefore (– 3) × (– 3) = 9 , (– 4) × (– 4) = 16 and so on. For negative root value ?-1 is i that is complex number.

In addition, negative numbers are not perfect squares and, therefore, have no square root in the system of rational numbers.

Square root is equal to the ’n’ power of 2

Example problems:

Example 1: How to Find the Square Root of a number 81?

Solution: ?81 = 9 x 9

?81 = 92

  = ?92

  = 9 is a perfect square

Example 2: Find the square root of a number 16.

16 = 2 x 2 x 2 x 2

  = 24

?16 = ?24  ( ? = power of 2)

  = 22

  =4 is the perfect square roots

The root of 16 is 4.