Tricks of square root

There are two types of numbers. Perfect square numbers and non-perfect square numbers. First we will understand the tricks for perfect square numbers then we will proceed to non-perfect square numbers.

A perfect square number can have only 0, 1, 4, 9, 6 and 5 as a unit digit.
unit digit of square root of a number

Tricks of square root for perfect square number:

Let us take a 4 digit number first and see the square root tricks related to it.

Example: 1764

Step 1: Focus on the unit digit of the number and identify the unit digit of square root.
In 1764 the unit digit is 4 so, unit digit of square root may be either 2 or 8.

Step 2: Make pairs of digits from the unit digit side.
1764 – Here 4 and 6 constitutes the 1st pair i.e. 64 whereas 7 and 1 form the 2nd pair i.e. 17.

Step 3: Find out the 2nd digit (tens digit) of square root with the help of 2nd pair.
Here the square root 17 lies between 4 and 5. Choose the smallest number. So, the 2nd digit will be 4 in square root of 1764.

Step 4: From the step 1 we know that unit digit may be either 2 or 8 and from the step 3 we know that tens digit is 4. It means possible numbers can be either 42 or 48. 

Step 5: To find whether the unit digit is 2 or 8. we need to consider tens digit and the next digit after tens digit (i.e. tens digit + 1) Multiplying these two terms:
4*(4+1) = 20 > 17
Here, 17 is smaller then 20, so square root of 1764 will be the smaller of the two options i.e. 42.
tricks of square root of a 4 digit numberExample 2: 7921

Step 1: unit digit of square root may be either 1 or 9.
Step 2: 21 forms the first pair whereas the 79 forms the 2nd pair.
Step 3: square root of 79 lies between 8 and 9. but 8 is the smaller one. so 8 will be the tens digit in square root of 7921.
Step 4: Possible numbers are 81 or 89. 
8*9 = 72 < 79
Here, 79 is larger than 72, so square root will be the larger of two options i.e. 89.
square root tricks for a 4 digit number

Square root tricks for a 5 digit number

Example: 13456

  1. Unit digit of the square root of 13456 can be either 4 or 6.
     Remaining digits constitutes the number 134. the square root of 134 lies between the 11 and 12. Here 11 is the smaller one so they will form the 2nd and 3rd digit of square root.
  2. Possible numbers are 114 and 116
    11*12 = 132 < 134, so square root of 13456 will be larger number from given options i.e. 116. 

Also know multiplication and square tricks

Non perfect square number

For non perfect square number, It is not possible to find exact value of square root with any square root trick but we can find a closer value to the right answer. For this we need to remember a simple formula. Which is written below.

    \[ \sqrt{X \mp Y} = \sqrt{X}\mp \frac{Y}{2\sqrt{X}} \]

Any number can be written as the addition or subtraction of two numbers X and Y, Where X is a complete square and Y can be any number added or subtracted to it.

Example: square root of 17

X = 16, Y=1

    \[ \sqrt{16 + 1} = \sqrt{16}+ \frac{1}{2\sqrt{16}} \]


    \[ = 4 + \frac{1}{2 * 4} \]


    \[ = 4 + 0.125 = 4.125\]

The actual value of square root of 17 is 4.123 which is very close to 4.125.

Also know: How to find the unit digit of a number raised to some power using the concept of cyclicity of numbers.

Example 2. 1613

We know that 1600 is a perfect square. So, X=1600, Y=13

    \[ \sqrt{1600 + 13} = \sqrt{1600}+ \frac{13}{2\sqrt{1600}} \]


    \[ = 40 + \frac{13}{80} \]


    \[ = 40 + 0.1625 = 40.162\]

What is perfect square number?
What is non perfect square number?


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