Square and Square Root Shortcut Tricks - Basic Math Tricks Part 5

Basic Tricks - PART- 5

Square & Square Root Shortcut Tricks

SQUARE up to 50

12 = 1	112= 121	212=441
22 = 4	122 = 144	222=484
32 = 9	132 = 169	232=529
42 = 16	142 = 196	242=576
52 =25	152 =225	252=625
62 = 36	162 = 256	262=676
72 = 49	172 = 289	272=729
82 = 64	182 = 324	282=784
92 = 81	192 = 361	292=841
102= 100	202 = 400	302=900

312= 961	412=1681
322= 1024	422=1764
332= 1089	432=1849
342= 1156	442=1936
352= 1225	452=2025
362=1296	462=2116
372=1369	472=2209
382= 1444	482=2304
392= 1521	492=2401
402= 1600	502=2500

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Square and Square Root of two digit get using formula1

Square and Square Root get using formula
Formula: (a+b)² = a²+2ab+b² i.e, (a / b)²= a² / 2ab / b²
we applied this formula to obtain the square of a number

Example 1:

( 57 )²
= ( 5 / 7 )²

Answer :
Apply formula of a²+2ab+b²
Consider,
57² = ?
A as 5
B as 7 (we break the number in two parts i.e, A as 5 and B as 7 and applied formula )
= 5² / 2 x 5 x 7 / 7²
= 25 / 2 x 5 x 7 / 49
a²= 25
b²= 49
2ab = 2 x 5 x 7 = 70
= 25 / 70 / 49
Step 1: Put down 9 carry 4
Step 2: add carry 4 to 70 = 74 put down 4 carry 7
Step 3: add carry 7 to 25 = 32 put down 32
and answer is 3249,
= 3249
All this do on your mind which will help in fast calculation to obtain the answer of Square and Square Root of a number.
we applied this formula to obtain the square of a number
This is similar to the above Example.

Example 2:

(69)²
= (6/9)²

Answer :
Consider A as 6, and B as 9.
= 6² / 2 x 6 x 9 / 9²
= 36 / 2 x 6 x 9 / 81 (we break the number in two parts i.e, A as 6 and B as 9 and applied formula )
a² = 36
b²= 81
2ab = 2 x 6 x 9 = 108
=36 / 108 / 81
Step1: put down 1 carry 8
Step2 : add 8 to 108 =116 then put down 6 carry 11
Step3 : and add 11 to 36 = 47 and put down 47
= so answer is 4761,
= 4761

Note: All this do on your mind which will help in fast calculation to obtain the answer of Square and Square Root of a number.

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Square and Square Root of three digit get using formula1

we applied this formula to obtain the square of three digit number

Example 1:

Square and Square Root of 114²
Answer :
Firstly we separate the 114 like this (11/4)²
then applied previous formula on it
=11² / 2x11x4 / 4²
=112 / 2x11x4 / 16
=121 / 88 / 16
=12996
we apply the formula a² + 2.a.b + b²
Step 1: note down 6 carry 1
Step 2: add carry 1 to 88 =  89, note down 9 carry 8.
Step 3: add carry 8 to 121 and note down 129
= 12996

Note: we can also separate 114 to find square like (1/14)2

Example 2:

Square and Square Root of 223²
Answer :
Firstly we separate the 223 like this(22/3)²
then applied previous formula on it
we apply the formula a² + 2.a.b + b²
= 22² + 2 x 22 x 3 + 3²
=484 / 132 / 9
= 49729
Step 1: note down 9
Step 2:note down 2 carry 13.
Step 3:add carry 13 to 4 = 17, note down 7
Step 4: add carry 1to 48 = 49 put down
= 49729
Note: All this do on your mind which will help in fast calculation to obtain the answer of Square and Square Root of a number.

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Square and Square Root of 100 base method

Example 1: 98²= ?

Answer :
Step 1: First we know that 98² is double of 98 that is = 98 x 98 = ?,
At first we count the number of less from 100. that is the above 98 is 2 less from 100.
Step2: Now we are going to multiply 2 x 2 = 4 and note down this 4 ( that are come from both less 98 x 98 from 100).
Step 2: put one Zero left from 4 and now subtract the less number is 2 from 98 that is = 96and the answer is 9604.

Example 2 : 96² = ?

Answer :
Step 1: First we know that 96² is double of 96 that is = 96 x 96 = ?,
At first we count the number of less from 100. that is the above 96 is 4 less from 100.
Step 2: Now we are going to multiply 4 x 4= 16 and note down this 16 ( that are come from both less 96 x 96 from 100).
Step 3: now subtract the less number is 4 from 96 that is = 92 and put down it
that is 9261
and the answer is 9216.

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Square and Square root a number ending in 6

Example 1:

76²
Step 1:put down 6
Step 2:Multiply 2 with (7 + 1) = 16 and add 16 +1 = 17.put down 7 and carry 1 .
Step 3:Multiply 7 with (7 + 1) = 56 + carry 1 = 57 put down 57
Answer is 5776

Example 2:

96²
Step 1:put down 6
Step 2:Multiply 2 with (9 + 1) = 20 and add 20 + 1 = 21.put down 1 and carry 2 .
Step 3:Multiply 9 with (9 + 1) = 90 + carry 2 = 92 put down 92
Answer is 9216

Example 3:

36²
Step 1:put down
Step 2:Multiply 2 with (3 + 1) = 8 and add 8 +1 = 9.put down 9.
Step 3:Multiply 3 with (3 + 1) = 12 put down 12
Answer is 1296

Example 4:

56²
Step 1:put down 6
Step 2:Multiply 2 with (5 + 1) = 12 and add 12 +1 = 13.put down 3 and carry 1 .
Step 3:Multiply 5 with (5 + 1) = 30 + carry 1 = 31 put down 31
Answer is 3136

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How to square a number ending with 5 within seconds

Squaring a number ending in 5 is the easiest if you know the trick.
Number given to you can be a two digit number or three digit number or five or six digit number, 

it does'n t matter.
The same trick can be applied to find the square of any number ending in 5.

Let us now discuss the steps to follow

Step 1:Multiply the ten's digit of the given number with its immediate next number.This will be the 1st digit of the answer.

Step 2:Put 25 next to the result of step 1.
Logic behind putting 25 is that the number will always end with 5 and 5² is 25. 

You get the answer

Let us find squares of numbers ending in 5 using this trick

Example 1:(35)²=?

Step 1: 3 is the ten's digit in the given number 35.
Immediate next number to 3 is 4.
Hence,by multiplying 3 & 4 we get 12(3x4=12).which is the 1st digit of the answer.

Step 2: Now putting 25 next to 12 we get the answer as 1225.

Answer: (35)²= 1225

Now you try to square 25,45,15,55,65,75,85,95 & see how quickly you can square.  

Example 2:(125)²=?

Step 1: 12 is the ten's digit in the given number 125.
Immediate next number to 12 is 13.
Hence,by multiplying 12 & 13 we get 156(12x13=156).which is the 1st digit of the answer.

[To multiply a 2digit by 2digit number when ten's digit of both numbers is 1 you can apply a trick to get your answer quickly.
1st digit =1 
middle digit =add unit's place digit of both numbers(2+3=5)
unit's place digit=Multiply unit place digit of both numbers(2x3=6)]

Step 2: Now putting 25 next to 156 we get the answer as 15625.

Ans (125)²= 15625

Now you try to square 105,115,135,145,155,165,175,185,195 & see how quickly you can square.  

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square number from 10 to 19 math trick

Easiest method to find the square of any number between 10 and 19 

Step 1: First add the given number and the units digit .

Step 2: Then Square the unit's digit and put the number next to the result obtained in STEP 1.

Ex-1:Find the square of 13 in 5 seconds

(13)² = ?

Step 1: 13 + 3 = 16_

Step 2: 3² = 9

put the result obtained in step 2 in step 1 we get

Ans=169

Ex-2:Find the square of 16 in 5 seconds

(16)² = ?

        Step 1: 16 + 6 = 22_

Step 2: 6² = 36

put the result obtained in step 2 in step 1 as shown below                                                                          

22_
 36

Ans=256

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Shortcut to square any number from(20-29)

Easy way to mentally square any number from 20 to 29

Step 1:First add the given number and the unit's digit.
Step 2:Double the result obtained in step 1.
Step 3:Square the unit's digit of the number given in question and place it next to the result obtained in step 2.

Example 1:Find the square of 22 in 5 seconds

            (22)²=?

             Step 1:22 + 2=24

             Step 2:24x2=48_

             Step 3:2²=4

Now put the result obtained in step 3 in step 2,we get

Ans:484 

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Shortcut to square any number from (30-79)

It is always better to have some maths tricks handy when you are planning to take any arithmetic aptitude test either in the competitive exams such as Bank exams,CAT,MAT.
Another easy trick to find square of the numbers between 30 to 79 is to take a common base as 50 and see how far the number is from 50.This method works well when the number is very close to 50.

Easy way to find the square of  number from 30 to 79

Step 1:Find how many more or less the given number is from 50.

Step 2:Add the number to 25 if more than 50 or subtract the number from 25 if less than 50.

Step 3:Then find the square of the number added or subtracted and put next to the result arrived at in step 2.

Let us now apply the trick that we learnt in the example below

Example 1:Find the square of 52 in 5 seconds

(52)²=?

Step 1: 52-50=2

           We find 52 is 2 more than 50

 Step 2: Here we notice that the given number is more than 50 so we add 25 as follows

            25+2=27

                                                                 
Step 3: Now we find the square  of 2
           2²=4

           putting the result obtained in step 3 next to the result obtained in step 2 after adding a 0 before it as it is a single digit,we get

           2704

Ans: (52)²=2704

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Shortcut to square any number from 90 to 99

This shortcut method is one of my favorite that I like to apply when I see a number such as 98 or 99 as all I need to know is how to subtract and square of numbers from 1 to 9.

Now lets go through the steps

Step 1:Assume 100 as base and find the difference between the number to be squared and 100.

Step 2:Subtract the difference you found in step 1 from the number to be squared to find tenth place digit.

Step 3:Square the difference and place it next to step 2.

Once you know the technique you can square any number from 90 to 99 mentally.

Let us see few examples to understand better

Example 1: 98²=?

Step 1:Assuming 100 as base, we shall find the difference
100-98=2
We get the difference as 2

Step 2:Subtracting the difference(2) from the number to be squared(98) we get,
98-2=96
96 is the tenth place digit

Step 3:Squaring the difference we found in step 1 we get
2²=4
4 is the Unit place digit.
98²=9604

Example 2: 99²=?

Step 1:Assuming 100 as base, we shall find the difference between the number to be squared(99) and 100.
100-99=1
We get the difference as 1

Step 2:Subtracting the difference(2) from the number to be squared(98) we get,
99-1=98
98 is the tenth place digit

Step 3:Squaring the difference we found in step 1 we get
1²=1
1 is the Unit place digit.
99²=9801

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Shortcut to square any number from 100-120

Base method is popularly used to find square of numbers when the numbers are close to a base.By choosing the right base you will arrive at your answer quickly.
So whenever you see 102,101,112,106 it should strike to you immediately .

Hey! this is quite close to 100 so let me take the base as 100 and see if I can get the answer soon.

Here we shall discuss the shortcut method to square any number from 100 to 120 using the base method.

Lets go through the steps now

Step 1:Take 100 as base and see how far the number is from 100.
Add the given number and its deviation.

Lets say you need to square 102.

Here you can notice that the number is 2 more than 100.
Adding the given number(102) and its deviation(2)
we get,

102+2=104

Step 2:Square the deviation(2)and place it next to the result obtained in step 1.
Squaring we get,
2²=4

Ans 10404

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Shortcut to find squareroot of any number

In every bank exam you are asked either to find the square root or cube root of a number. By knowing the shortcut to find the square root of a number, you will be able to find out the square root of any number within seconds.  

Now lets go through the steps...

Step 1: First of all group the number in pairs of 2 starting from the right.

Step 2: To get the ten’s place digit, Find the nearest square (equivalent or greater than or less than) to the first grouped pair from left and put the square root of the square.

Step 3: To get the unit’s place digit of the square root

Remember the following

If number ends in	Unit’s place digit of the square root
1	1 or 9(10-1)
4	2 or 8(10-2)
9	3 or 7(10-3)
6	4or 6(10-4)
5	5
0	0

Lets see the logic behind this for a better understanding

We know,

1²=1

2²=4

3²=9

4²=16

5²=25

6²=36

7²=49

8²=64

9²=81

10²=100

Now, observe the unit’s place digit of all the squares.

Do you find anything common?

We notice that,

Unit’s place digit of both 1² and 9² is 1.

Unit’s place digit of both 2² and 8² is 4

Unit’s place digit of both 3² and 7² is 9

Unit’s place digit of both 4² and 6² is 6.

Step 4: Multiply the ten’s place digit (found in step 1) with its consecutive number and compare the result obtained with the first pair of the original number from left.

Remember,

 If first pair of the original number > Result obtained on multiplication then  select the greater number  out of the two numbers as the unit’s place digit of the square root.

If firstpair of the original number < the result obtained on multiplication,then select the lesser number out of the two numbers as the unit’s place digit of the square root.

Let us consider an example to get a better understanding of the method

Example 1: √784=?

Step 1: We start by grouping the numbers in pairs of two from right as follows

7 84

Step 2: To get the ten’s place digit,

We find that nearest square to first group (7) is 4 and √4=2

Therefore ten’s place digit=2

Step 3: To get the unit’s place digit,

We notice that the number ends with 4, So the unit’s place digit of the square root should be either 2 or 8(Refer table).

Step 4: Multiplying the ten’s place digit of the square root that we arrived at in step 1(2) and its consecutive number(3) we get,

2x3=6

ten’s place digit of original number > Multiplication result
7>6

So we need to select the greater number (8) as the unit’s place digit of the square root.

Unit's place digit =8

Ans:√784=28