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## Square of any number by easy Vedic math’s trick

It is really a lengthy & complicated to find Square & Square-root of any number. Here we apply method which is very easy & fast to find the square of any number. This method is work like Base method of multiplication.

To find square of any number Vedic sutra 13 Sopaantyadvayamantyam is applicable.

#### Case 1: Number is near the power of 10 like 10,100,1000 etc.

Ex.i) 82

Here, Base = 10   (power of 10 number as base)

Difference = Number – Base

= 8-10 = -2

If we divide our answer in two parts as, Put the values in formula

RHS = (-2)2 = 4

Note: Number of digit in RHS is equal to number of Zero in base. If less digit is there add 0 in left side of RHS & if number of digits are more then carryover to extra digit in LHS of the answer. Put the values in formula,

LHS = 8+(-2) =6

Ex. ii) 932

Here, Base = 100 (As number is closed to 100)

Difference = 93-100 =-7

RHS = Difference2= (-7)2 = 49

LHS = Number + Difference

= 93 + (-7) = 86

Ex. iii) 9922

Here base = 1000

Difference = 992 – 1000 =- 008

= (992+(-008)) /(-008)2

= 984/064  (We add zero as base is 1000, so 3- digit should be there)

Similarly,

Ex.iv)1122

Base = 100 & Difference = 112-100 = 12

= 124/144    (As base is 100 so, only 2-digit in RHS & carryover 1)

Case 2: Number is NOT near the power of 10 like 10,100,1000 etc.

If number is not near the 10, 100 1000 etc. in such case we consider 2- base one is Actual base & other is working base. Actual base is the base we used in above sum where as Working base is the number which is multiplier of 10 like 30,50,340,400,550 etc.

e.g. Number = 712

Here Actual base = 1000 (As number is close to 1000)

difference = 1000-712 = 288,

which is a big number & it is complicated to find difference2 .  So, here we consider working base the number which is near the given example & also multiplier of the 10.

Here working base = 700.

Ex. v) 3922

Actual base = 100

Working base = 400 = 4 x Actual base

Difference = 392-400= (-08)

RHS = Diff.2=(-08)2=64

LHS = Num. + Diff. =392 + (-8) = 384

But working base = 4 x Actual base

So Actual LHS = 4 x LHS

= 4 x 384 =1536

vi) 672 =

Actual base = 10

Working base = 70 = 7 x Actual base

Difference = 67-70= (-3)

RHS = Diff.2=(-3)2=9

LHS = Num. + Diff. =67 + (-3) = 64

But working base = 7 x Actual base

So Actual LHS = 7 x LHS

= 7 x 64 =448

Ex. vii) 5132

Actual base = 100

Working base = 500 = 5 x Actual base

Difference = 513-500= 13

RHS = Diff.2=(13)2=169

LHS = Num. + Diff. =513 + 13 = 526

But working base = 5 x Actual base

So Actual LHS = 5 x LHS

= 5 x 526 =2630

Answer, 5132 = 2630/169     (As base is 100 so, only 2- digit should be in RHS  & carryover 1)

Similarly,

Ex. viii) 382

Actual Base = 10

Working Base =40 =4 x 10

=36 x 4/4

Ex. ix) 1032      Base = 100

103+03/32 =10609       (add 0 in RHS, as base is 100)

Ex. x) 1472

Actual Base = 100

Working Base =150 =(3/2) x 100

=144  x (3/2)/ 09

=21609

OR

Actual Base = 10

Working Base =150 =15 x 10

=144 x 15/ 9

=21609

## Square-root of perfect square

As most of the schools & colleges ask for square-root of perfect square number so we find same here. For imperfect number it is little bit complicated so, we will study it in advanced level.

First, we memorized the square of the number 1 to 10 By observing last digit of square, we form one more table. After observing the above table we can conclude that in the column of last digit of square number 2,3,7 & 8 are absent. That means the number having last digit 2,3,7 & 8 are not a perfect square.

Now   to simplify our calculation we convert our table 1 as: By using above table, we can find out square root of number up to 10000.

Ex. i) Find the square root of 8464

Step 1: Check the last digit of given square.

Here last digit is 4 so, from table 2 the last digit of the square root will be 2 or 8.

Step 2:  Check the approximate square-root from table 3

From Table 3 we can conclude that number 8464 is lies between It means our square-root is lies in between 90 & 100.

Step 3:  From step 1 we know that last digit of square-root is 2 or 8, so from  step 1 & 2 we can say that answer will be 92 or 98.

Step 4: Observe that number 8464 is close to 8100 or 10,000.

Here it is close to smaller number 8100, so our answer is also smaller number i.e. 92. So our answer is 92.

Answer, Square-root of 8464 = 92.

Ex. ii) Find the square-root of 2209

Step 1: Check the last digit of given square.

Here last digit is 9 so, from table 2 the last digit of the square root will be 3 or 7.

Step 2:  Check the approximate square-root from table 3

From Table 3 we can conclude that number 2209 is lies between It means our answer is lies in between 40 & 50.

Step 3:  From step 1 we know that last digit of square-root is 3 or 7, so from step 1 & 2 we can say that answer will be  43 or 47.

Step 4: Observe that number 2209 is close to 1600 or 2500.

Here it is close to bigger number 2500, so our answer is also bigger number i.e. 47.

Answer, Square-root of 2209 = 47.

Now to find the square -root of five digit number we need to memorized square of 11 to 20 & convert in to 10’s multiplier. Ex iii) Find the square-root of 16129.

Step 1: As last digit of number is 9,

so last digit of answer will be 3 or 7 from table 2.

Step 2: Observe table 4 16129 is between the square of 120 & 130.

Step 3: From step 1 last digit of answer is 3 or 7 means

answer will be 123 or 127.

Step 4: 16129 is closer to 16900 means square root is also a bigger number, i.e. 127.

Answer, Square- root of 16129= 127.

Similarly,

Ex. iv) Square-root of 33856

From table 2 last digit of answer will be 4 or 6.

Answer is in between 180 & 190 from table 4.

Answer will be 184 or 186, from above 2 steps

From table 4 Number 33856 is close to 32400, which is smaller number

So, answer is also smaller number i.e. 184.

Answer, Square-root of 33856 = 184.

Ex. v) Square-root of 7225

From table 2 last digit of answer will be 5.

From table 4 answer is in between 6400 & 8100, so our answer will be in  between 80 & 90 from table 4.

Answer, Square-root of 7225 = 85.

Try This:

i) 1072   ii)9892  iii) 1672  iv) 11202

Find the square-root of,

v)8649   vi)30,276   vii) 3136

Find the below link to study more:

3. Vinculum & De-vinculum of Number

4. Base Method of Multiplication 