Properties of Square Roots with Examples

In this section, we will discuss all properties of square roots.

Property 1

If a number is a perfect square, then its square root will be a whole number. For example, we know that 100 is a perfect square number. Its square root √100=10 is a whole number.

More examples of perfect squares: $4, 9, 16, 25, 36, 49, 64, 81$ etc.

Property 2

The square root is denoted by the radical symbol $\sqrt{}.$ If $x$ is a non-perfect number, then its square root $\sqrt{x}$ is a quadratic surd. In this case, $\sqrt{x}$ will be an irrational number.

For example, as $11$ is not a perfect square, $\sqrt{11}$ represents an irrational number.

Property 3

The square root can be expressed as power 1/2. So the square root of a number $x$ is written as $x^{1/2}$or $x^{0.5}$ in exponential form (Laws of Exponents).

For example, $5^{1/2}$ or $5^{0.5}$ is the exponential form of the square root of $5.$

Property 4

If the square root of a number is an integer, then the integer has the last digit $0, 1, 4, 5$ and $9.$ Thus if an integer is the square root of some numbers, then the integer cannot have the last digit $2, 3, 6, 7$ or $8.$

For example: √100=10, √121=11, √16=4, √25=5, √81=9. Note that all these square roots end with the digits $0, 1, 4, 5$ or $9.$

Property 5

If the last digit of a number is either $2, 3, 7$ or $8,$ then the square root of that number cannot be a whole number. Thus, the square root of any even number with last digit 2 or 8 is an integer.

For example, the square roots of the numbers 22, 228, 888, 772 are not integers.

Property 6

As $0^2=0,$ $1^2=1,$ $2^2=4,$ $3^2=9,$ $4^2=16,$ $5^2=25,$ $6^2=36,$ $7^2=49,$ $8^2=64$ and $9^2=81$, we deduce the following Important Properties for the square root of a number:

1. If a number has last digit $0,$ then its square root will have the last digit $0.$

2. If a number has last digit $1,$ then its square root will have the last digit either $1$ or $9.$

3. If a number has last digit $4,$ then its square root will have the last digit $2.$

4. If a number has last digit $5,$ then its square root will have the last digit $5.$

5. If a number has last digit $6,$ then its square root will have the last digit either $4$ or $6.$

6. If a number has last digit $9,$ then its square root will have the last digit either $3$ or $7.$

Property 7

The square root of an even number is always even. For example, $\sqrt{64}=8.$ Note that both $8$ and $64$ are even numbers.

Property 8

The square root of an odd number is always odd. For example, √121=11. Note that both 11 and 121 are odd numbers.

Property 9

If a number ends with an even number of zeros, then the square root of that number will be an integer.

For example, 100 has 2 zeros at the end and √100=10 is an integer. Similarly, the number 10000 ends with four zeros, and its square root √10000=100 is an integer.

Property 10

If a number ends with an odd number of zeros, then the square root of that number will not be a whole number. As a result, we can say that the square root of a number with an odd number of zeros at the end will be an irrational number.

For example, 10 has only one zero at the end and √10 is not a rational number. Similarly, the number 1000 ends with three zeros, and its square root √1000 is also an irrational number.

Property 11

If $x$ is a square root of a number X, then $-x$ is also a square root of the number $X.$ More specifically, we have \[\sqrt{X}=\pm x.\]

Property 12

The square root of a negative number does not exist in the set of real numbers. But they do exist in the set of complex numbers. Note that the square root of a negative number is called a purely imaginary number. For example,

$\sqrt{-7}=\sqrt{7 \times -1}$ $=\sqrt{7} \times \sqrt{-1}$ $=7i,$

where $i=\sqrt{-1}$ (a square root of 1).

Property 13

For a natural number $n,$ the square $n^2$ is equal to the sum of first $n$ odd numbers. Mathematically, 1+3+5+…+(2n-1)=n2, that is, the sum of first n odd natural numbers is n2.

For example, 25 = 52 = 1+3+5+7+9 = the sum of first 5 odd numbers.

Property 14

Multiplication rule for square roots: For two numbers $x$ and $y,$ we have \[\sqrt{x \times y}=\sqrt{x} \times \sqrt{y}.\]

This shows that the square root of the product of two numbers is equal to the product of the square roots of those two numbers. The same rule holds for more than two numbers.

For example, $\sqrt{8}=\sqrt{4 \times 2}$ $=\sqrt{4} \times \sqrt{2}$

Property 15

Division rule for square roots: For two numbers $x$ and $y$ with $y \neq 0,$ we have \[\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}.\]

∴ the quotient of the square roots of two numbers is the same as the square root of the quotient of those two numbers.

For example, $2=\sqrt{\frac{8}{2}}=\frac{\sqrt{8}}{\sqrt{2}}$