** 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) 8 ^{2 }**

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

**Difference = Number – Base**

= 8-10 = -2

If we divide our answer in two parts as,

Answer = LHS/RHS

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

**Answer** = 6/4 =64

**Ex. ii) 93 ^{2}**

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

Difference = 93-100 =-7

**RHS** = Difference^{2}= (-7)^{2} = 49

**LHS** = Number + Difference

= 93 + (-7) = 86

**Answer,** 93^{2} = 8649

**Ex. iii) 992 ^{2}**

^{ }**Here base = 1000**

Difference = 992 – 1000 =- 008

Answer = (Number+Difference)/Difference^{2}

= (992+(-008)) /(-008)^{2}

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

**Answer**, 992^{2} = 984064

**Similarly**,

**Ex.iv)112**^{2 }

^{ }Base = 100 & Difference = 112-100 = 12

Answer = (112 +12)/12^{2}

^{ }= 124/_{1}44 (As base is 100 so, only 2-digit in RHS & carryover 1)

**Answer**, 112^{2} = 12544

**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 difference^{2} . 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) 392 ^{2}**

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

** Answer**, 392^{2 }=1536/64 = 153664

**vi) 67 ^{2} =**

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

**Answer**, 392^{2} =448/9 = 4489

**Ex. vii) 513 ^{2}**

** **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, 513^{2 }= 2630/169 (As base is 100 so, only 2- digit should be in RHS & carryover 1)

** Answer**, 513^{2 }= 263169

**Similarly,**

**Ex. viii) 38 ^{2 } **

Actual Base = 10

**Working Base =40 =4 x 10**

Answer =38+(-2) x 4/(-2)^{2 }

=36 x 4/4

**Answer,** 38^{2 }=1444

** **

**Ex. ix) 103 ^{2 }Base = 100**

** **103+03/3^{2 }=10609 (add 0 in RHS, as base is 100)

** **

**Ex. x) 147 ^{2 }**

**Actual Base = 100 **

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

** ^{ Answer }**= 147+(-3)x(3/2)/(-3)

^{2}

^{ }

=144 x (3/2)/ 09

=21609

** OR**

^{ }**Actual Base = 10**

**Working Base =150 =15 x 10 ^{ }**

** ^{ Answer }**= 147+(-3) x 15/(-3)

^{2}

^{ }

^{ }

=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.

So, answer will be 85.

**Answer, Square-root of 7225 = 85.**

**Try This:**

i) 107^{2 } ii)989^{2} iii) 167^{2} iv) 1120^{2}

Find the square-root of,

v)8649 vi)30,276 vii) 3136

**Find the below link to study more:**

1.Find the cube & cube-root of any number by just observing it

2. Easiest way to find the Square of number ends with 5.

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