In mathematics, all real numbers consist of two sets of square root, one is a positive square root and another is a negative square root. The square root of a positive number can also be called the principal square root. When we multiply any two positive numbers, we get a positive product as a result because both the positive numbers that we have multiplied have the same sign. In the same way, if we multiply two negative numbers, we get a negative product. But the real question is what is a square and a square root? Well, if a number is multiplied by itself to generate a product, the resultant product will be called the square of that number. For example, 0.4 x 0.4 = 0.16. Here 0.16 is the square of 0.4. Square root can be defined as a number that multiplies itself to give the original number. For example, let us take a number 9. The square root of 9 will be 3 because it multiplies itself twice to give 9 i.e., 3 x 3 = 9.

The square root of 289 is 17. It can also be written as:

There are five methods with the help of which we can find the square root of any number. So let us learn about them.

The first method is the Prime Factorization Method

The second method is the Repeated Subtraction Method

The third is the Long Division Method

The fourth method is the Number Line Method

And the last method is known as the Average Method

The perfect square exists only with the perfect square.

The square of an even perfect square will always be even whereas the square of an odd perfect square will always be odd.

In a negative number, the square root cannot be defined.

A number that is ending with 1, 4, 5, 6, or 9 will always have a perfect square root.

The perfect square root is not possible with numbers ending with 2, 3, 7, or 8.

There can be no square root if the number is ending with the odd number of zeros.

When we are dealing with large numbers; we often use the division method to find the square root. The steps to use the division method are given below.

Step 1) Mark the digits in pairs and always start with the unit digit. Every pair and remaining one digit is known as a period.

Step 2) Think of the largest number you can think whose square will be equal to or maybe just less than the first period.

Step 3) Consider the number as the divisor as well as the quotient.

Step 4) The step will be to subtract the product of the divisor and the quotient from the first period and then bring the next period down to the right of the remainder. This will become the new dividend.

Step 5) Finally, a new divisor will be obtained by taking the quotient twice and annexing with it a suitable digit which we can also take as the next digit of the quotient, selection should be in such a way that the product of new divisor and this digit must be equal to or just less than the new dividend.

Example 1) Find the value of 7√1 - 5√1 + 2√1 using the value of under root 1.

Solution 1) Value of √1 = 1

=7√1 - 5√1 + 2√1

= 7 (1) - 5 (1) + 2 (1)

= 7 - 5 + 2 = 4.

Example 2) Solve for p if p2 + 8 = 3

Solution 2) p2 + 8 = 3

p2 = 3 - 8

p2 = - 5

p = √-5 = √-1 . √5

p = √5i

FAQ (Frequently Asked Questions)

1. What is the Difference Between Square Root and Under Root

We can explain the difference between a square root and an under root by comparing both of them. We can use it under root for multiple purposes. For example, we can use under root for the square root, cube root, and also the nth root. In fact, it wouldn’t be wrong to say that under root is just another way we can express the nth root. On the other hand, if we talk about square root, it can be used for a square root only.

For example, 25 under root to power 2 is the square root of 25, which is 5. But on the other hand, 5 is the square root of 25.

2. Can we find Square Roots Using the Repeated Subtraction Method?

One of the square root properties says that if a natural number is a square number, it will definitely be the sum of successive odd numbers starting from 1. So we will test this property and find out the square root of a number using the repeated subtraction method. Now the question is how this method works? We can find the square root of a number if we repeatedly subtract the successive odd numbers from the given square number, till it reaches zero.

Here is an example: let us find the square root of 81 with the help of the repeated subtraction method.

81 - 1 = 80

80 - 3 = 77

77 - 5 = 72

72 - 7 = 65

65 - 9 = 56

56 - 11 = 45

45 - 13 = 32

32 - 15 = 17

17 - 17 = 0

0 comes in the 9th step, therefore, the square root of 81 will be 9 i.e √81= 9.