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.
Now we shall learn few basic Vedic mathematics methods or Vedic Maths Squaring Tricks:
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.
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:
Square root Tricks! Learn how to find square roots
The square root of a number is the inverse operation of square of number.
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 
 Observing the square of numbers from 1 to 9, we can analyze that an exact square ends with 1,4,5, 6,9, and 0 square root. These numbers will be rational numbers.

 If the square of the number ends with 2,3,7 and 8 then its square root will be an irrational number.

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
\(\begin{align}\sqrt {2} = 1 + 0.5 \end{align}\)
\(\begin{align}\sqrt {2} = 1.5\end{align}\)
External References
Wikipedia.org  Vedic Mathematics