Table of Contents
1.  Introduction 
2.  Squaring Tricks! Square numbers faster 
3.  Square root Tricks! Learn how to find square roots 
4.  Summary 
5.  FAQs 
12 November 2020
Reading Time: 10 Minutes
Introduction
Do you know how to square big numbers?
Do you find it hard or take too long to square numbers?
Bharthi Krishna Tritha, an Indian monk, wrote the book "VEDIC MATHEMATICS", which has a list of mathematical tricks to solve maths calculations faster than traditional methods. It gives the easiest way to solve addition, subtraction, multiplication, division, and related problems. Vedic mathematics can applied to pneumatics, statics, astronomy, and the financial domain.
Here we will learn about how to find the square root of a number.
Also read:
 Vedic Math Examples
 Vedic Math Features
 Addition and Subtraction Tricks
 Multiplication and Division Tricks
 Vedic Math Tricks and their Importance
Squaring Tricks! Square numbers faster
A number, when multiplied by the number itself the product obtained, is called "SQUARE OF THAT NUMBER ". Here are some examples followed by Square tricks.
Example :
Number  Square of the number 
6^{2}  6 × 6 = 36 
8^{2}  8 × 8 = 64 
12^{2}  12 × 12 = 144 
Now we shall learn few basic Vedic mathematics methods or Vedic Maths Squaring Tricks:
Type 1: Squaring of number ending with 5.
Sutra is "BY ONE MORE THAN PREVIOUS ONE"
Step 1: Add 1 to the first digit from the left and multiply by the number itself.
Step 2: Add 52 (25) at the end to the number obtained from step 1.
Number  Steps 
65  
6 × (6 + 1) = 6 × 7 = 42 
Add 1 to the left number (6) and multiply by the number itself and 
=4225  Add 5^{2} (25) at the last of 42 
Answer : 65^{2} = 65 × 65 = 4225  
85  
8 × (8 + 1) = 8 × 9 = 72 
Add 1 to the left number (8) and multiply by the number itself and 
= 7225  add 5^{2} (25) at last of 72 
Answer : 85^{2} = 85 × 85 = 7225  
155  
15 × (15 + 1) = 15 × 16 = 240 
Add 1 to the left number (15) and multiply by the number itself and 
= 24025  add 5^{2} (25) at last of 240 
Answer : 155^{2} = 155 × 155 = 2402 
Type 2: Squaring of numbers less than 50 and numbers not ending with 5.
Number  Steps 
34  50  16 = 34 
5^{2} = 25 = 25 + (16) = 9 
Square the first digit (5) of first part (50) then add part (16) 
16^{2} = 256  Square the second part of number (16) 
9 + 256 = 1156  Add the answers got in step 1 (9)and step 2 (256) 
Answer : 34^{2} = 34 × 34 = 1156  
28  50  22 = 28 
5^{2} = 25 = 25 + (22) = 3 
Square the first digit (5) of first part (50) then add second part (22) 
22^{2} = 484  Square the second part of number (22) 
3 + 484 = 784  Add the answers got in step 1 (3)and step 2 (484) 
Answer : 28^{2} = 28 × 28 = 784 
Type 3: Squaring of numbers less than 50 and numbers not ending with 5.
Number  Steps 
74  50 + 24 = 74 
5^{2} = 25  Square the first digit (5) of first part (50) then add 
= 25 + 24 = 49 
second part (24) 
24^{2} = 576  Square the second part of number 24 
49 + 576 = 5476  Add the answers got in step 1 (49)and step 2 (576) 
Answer : 74^{2} = 74 × 74 = 5476  
57  50 + 7 = 57 
5^{2} = 25 = 25 + 7 = 32 
Square the first digit (5) of first part (50) then add second part 7 
7^{2 }= 49  Square the second part of number 7 
32 + 49 = 3249  Add the answers got in step 1 (32)and step 2 (49) 
Answer : 57^{2} = 57 × 57 = 3249 
Type 4: Squaring of number near to their base 10,100,1000, and so on:
Number  Steps 
105  100 + 5 = Divide the given number to their base and number 
105 + 5 = 110  Add the second part of number 5 to the given number (105) 
5^{2} = 25  Square the second part of the 5^{2} 
11025  Combine the numbers from step 1 and step 2 
Answer : 105^{2} = 105 × 105 = 11025  
986  1000  986 = 14 
986  14 = 972 
The given number 986 is less than 14 from its base value 1000, so the deficient number 14 should be subtracted by the given number 986 
14^{2} = 196  Square of deficient number 211 
972196  Combine the numbers from step 1 and step 2 
Answer : 986^{2} = 986 × 986 = 972196 
 If the number is lesser than its nearest base number then the deficient number is reduced from the given number.
 If the given number is greater than its nearest base number then the surplus number is added to the given number.
Type 5: Squaring of a number near to their sub base:
Number  Steps 
306  300 + 6 = Divide the given number to their sub base and number 
3 × (306 + 6) = 3 ×312 
Add the second part of number 6 to the given number (306) and multiply it by 3 
6^{2 }= 36  Square the second part of the 6^{2} 
93636  Combine the numbers from step 1 and step 2 
Answer : 306^{2} = 306 × 306 = 93636  
480  500  480 = 20 
480  20= 5 ×(480 20) =5 × 460 = 2300 
The given number 480 is less than 20 from its sub base value 500, so the deficient number 20 should be subtracted by the given number 480 and multiplied by 5 
20^{2} =400  Square of deficient number 211 
230400  Combine the numbers from step 1 and step 2 
Answer : 480^{2} = 480 × 480 = 230400 
GENERAL METHOD TO FIND SQUARE OF NUMBER OR DUPLEX METHOD :
Duplex combination or Dwanda yoga is term used in terms for squaring and multiplication,denoted as D. Following are the basic methods used in duplex:
 D (a) = a^{2}
 D (ab) = 2(ab)
 D (abc) = 2 (ac) + b^{2}
 D (abcd) = 2 (ad) + 2(bc)
 D (abcde) = 2 (ae) + 2(bd) + c^{2}
Example :
Square root Tricks! Learn how to find square root
The square root of a number is the inverse operation of square of number.
Example
Number  Square of the number  Square root of the number 
6^{2}  6 × 6 = 36  \(\begin{align}\sqrt {36} = 6\end{align}\) 
8^{2}  8 × 8 = 64  \(\begin{align}\sqrt {64} = 8\end{align}\) 
12^{2}  12 × 12 = 144  \(\begin{align}\sqrt {144} = 12\end{align}\) 
Points to be remembered before finding the square root of any number :
 The given number is arranged as a group of two numbers from right to left. If a single number has remained at the left, then it is also considered as one group.
 The number of groups derived for the given number will be the number of digits of the square root of that number.
Example : \(\begin{align}\sqrt {25} = 5\end{align}\), here there are two digits so one group. So, the square root of 25 is 5.
\(\begin{align}\sqrt {169} = 13\end{align}\), the number forms two groups (1, 69), so the square root is twodigit number 13.
3. If the square of a number is an odd number, then the square root is also an odd number, and if the square is an even number, then the square root is also an even number.
4.
Number  Square 
1^{2}  1 
2^{2}  4 
3^{2}  9 
4^{2}  16 
5^{2}  25 
6^{2}  36 
7^{2}  49 
8^{2}  64 
9^{2}  81 
Logic to find the square root 


Square Number ending with following number  Square root ends with following number 
1  1 or 9 
4  2 or 8 
5  5 
6  4 or 6 
9  3 or 7 
Sutra is "FIRST BY THE FIRST AND LAST BY THE LAST"
Let's solve few examples and learn how this technique works :
Type 1: Square root of perfect square number:
Number:\(\begin{align}\sqrt {3969}\end{align}\)  Step 
39  69  Grouping: from right to left 
38 is greater than 36, the perfect square number of 62, and 69 is lesser than 64, which is a perfect square number of 82  
3600  6400 602  802 
Therefore the given number lies between 3600 and 6400 that means there square number will be between 60 and 80 
So the first number in the square root will be 6  
The number in the units places is 9 so the square number should be ending with 3 or 7 (as discussed above)  
The square root of the given number should be either 63 or 67  
Applying digit sum method to find the square root  
63 = 3969 (6 + 3) = 3+9+6+9 9 = 2 + 7 9 = 9 
So the square root of number 3969 is 63 
Answer : 3969 = 63 
Type 2: Square root of imperfect square number :
Number  Steps 
\(\begin{align}\sqrt {2}\end{align}\)  The given number is not a perfect square 
1 + 1 = 2. or 4  2 = 2 
The nearest square number is 1 and 4 
1 + 1/2 1 + 0.5 1.5 
Square root of 1 is 1, add (1 ÷ [ 1 + 1]) i.e 1 divide by twice the perfect square number 
1.5  The same when calculated in calculator gives 1.414 which is approximately equal to 1.5 
\(\begin{align}\sqrt {20}\end{align}\)  \(\begin{align}\sqrt {2}\end{align}\) 
25  5 = 20  25 is the perfect square of 5 subtract it from (1/10) 1 divide by twice the perfect square 
25  5/10 5  0.5 4.5 
The same, when calculated in calculator, gives 4.47 which is approximately equal to 4.5 
Summary
After learning a few of the tricks to find square and square roots of the number, one will be able to easily find the square numbers and their square roots without any help from calculators. These tricks are very helpful in solving aptitude problems related to square and square roots in competitive exams.
Written by Nethravati C, Cuemath Teacher
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FAQs
What is the square of a number?
If a number is multiplied by the same number itself then it is called a square of the number.
Example: square of 6 = 6^{2}
= 6 × 6 = 36
What is the square root of a number?
A number which when multiplied by the number itself gives a square of that number the inverse operation of this is known as the square root of the number.
Example \(\begin{align}\sqrt {25}=5\end{align}\)
How to find the square root of a number?
Step 1 : group the given number as two digits in one group from right to left.
Step 2 : find the nearest perfect square number of the grouped numbers and analyse between which numbers the given number lies.
Step 3 : depending on the number in units place decide which number will be the possible square number
Step 4 : apply the digit sum method to find the possible square number.
How to find the square of a number?
Square of a number can be determined by multiplying the number by the same number .
Example: 16^{2} = 16 × 16 = 256
What is the square root of 5?
The nearest perfect square number is 4, so 5 can be written has
\(\begin{align}\sqrt {5}=\sqrt {4}+1\end{align}\)
Adding two times of 4 i.e 8 to the denominator of second part
\(\begin{align}\sqrt {5}=\sqrt {4}+1/8\end{align}\)
Simplifying
\(\begin{align}\sqrt {5}=2 + 0.125\end{align}\)
\(\begin{align}\sqrt {5}= 2.125\end{align}\)
What is the square root of 3?
The nearest perfect square number is 4, so 3 can be written has
\(\begin{align}\sqrt {3}=\sqrt {4}1\end{align}\)
Adding two times of 4 i.e 8 to the denominator of second part
\(\begin{align}\sqrt {3}=\sqrt {4}1/8\end{align}\)
On simplifying
\(\begin{align}\sqrt {3}=2  0.125\end{align}\)
\(\begin{align}\sqrt {3} = 1.875\end{align}\)
What is the square root of 8?
The nearest perfect square number is 9, so 8 6can be written has
\(\begin{align}\sqrt {8} = \sqrt {9}1\end{align}\)
Adding two times of 9 i.e 18 to the denominator of second part
\(\begin{align}\sqrt {8} = \sqrt {9}1/18\end{align}\)
On simplifying
\(\begin{align}\sqrt {8} = 3  0.055 \end{align}\)
\(\begin{align}\sqrt {8} = 2.945 \end{align}\)
What is the square root of 2?
The nearest perfect square number is 1, so 2 can be written has
\(\begin{align}\sqrt {2} = \sqrt {1}+1\end{align}\)
Adding two times of 1 i.e 2 to the denominator of second part
\(\begin{align}\sqrt {2} = \sqrt {1}+1/2\end{align}\)
On simplifying
\(\begin{align}\sqrt {2} = 1 + 0.5 \end{align}\)
\(\begin{align}\sqrt {2} = 1.5\end{align}\)