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Square Root of Imperfect Square

We already study how to find the square root of perfect square. Now here we study the method to find the square root of imperfect square.

The method used for perfect Square number is very easy and students can find it very useful.

Where as the method used for imperfect square number is slightly difficult, but once you understand the steps, it will be easy for you to find the square root of any number.  This is very useful for the students appearing for competitive exam & professionals.

There are different methods to find the square root of general number(perfect as well as imperfect). We will study both Vedic Method & the method explained by some mathematicians.

Method 1:

This is easiest method among all to solve the square root of imperfect square.  But this method is applicable for small  numbers, as we need to divide the given number in to two parts.(As we should know the nearer perfect square number)

Where as first part is the nearer perfect square number & second part is the remaining number.

e.g. ii) Find Square root of 38.

Now in given number the nearer perfect square number is 36.

So, we can write the given number as, 36 + 2,

Say a =  Nearer Perfect Square number = 36

b = Difference = 38 -36 =2.

In general form we can write,

Now we put the given number in formula,

Put the Values in given formula,

= 36  + (2/2*√36) (We use + sign as number divide in (a + b) form)

= 6 + (1/6)

=37/6 = 6.16

Ans, square root of 38 = 6.16

By using only one formula we get the answer.

This is easiest & fastest method of find the square root of any number.

To understand the method, we will be solving some more examples.

e.g ii) Find the square root of 62

Now in this sum nearer perfect square number is 64.

So, we can write it as,

64-2

Here, a = 64 & b = 2

Now put the values in formula,

= 64 – (2/2*64)    (We use – sign as number divide in (a – b) form)

= 8 – (1/8)

= 63/8 = 7.87

Ans, Square root of 62 = 7. 87

e.g. iii) Find the square root of  634

Now here nearer perfect square number is 625.(25)

So, we can write our number as,

625 + 9

Now, put the values in formula,

= 625 + 9/2*625

= 25 + 9/50

= 1259/50 =25.18

Ans, Square root of 634 = 25.18

Method 2:

This method is lengthy than method 1, but you can find square root of any number without knowing the nearer perfect square.

e.g. iv) Find the square root of 784

Step 1: Very first step while calculating Square root is divide the given number in to groups of two digits,  starting from right. At the end if single digit is left, consider it as a one more group.

Here, 784 = 7   84

Start from right, take last two digits 84,  now last 7 is remaining, consider it as separate group.

Step 2: We will start with first group i.e. 7, try to find the perfect square number just smaller than 7. Here, the number is 4, and we get it by, 2 x 2.

Hence, we put 2 in divisor column & other 2 in quotient column. Write their product 4, below 7 and do subtraction.

Now onward after every step,  we get new divisor by doing addition of new Quotient & divisor

So, our new divisor is, 2(Quotient) + 2(Divisor) = 4

And bring the next group 84, down. Now our dividend = 384.

Step 3: Here apply the rule that, A new divisor can be multiplied by only that number which is suffixed to it. In this case

1. If we suffix 1 to 4 it will be, 41 & 41 x 1 = 41
2. If we suffix 2 to 4 it will be, 42 & 42 x 2 = 84

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1. If we suffix 8 to 4 it will be 48, 48 x 8 = 384

So, here we get our answer, as after putting this value remainder is zero.

Ans, Square root of 784 = 28.

e.g. v) Find the square root of  1034

Step 1: Divide the given number in to groups of two digits,  starting from right. At the end if single digit is left, consider it as a one more group.

Here, 1034 = 10  34

Start from right, take last two digits 34, & next group is 10.

Step 2: We will start with first group i.e. 10, Here 9 is a  perfect square number just smaller than 10. Obtained by, 3 x 3.

Hence, we put 3 in divisor column & other 3 in quotient column. Write their product 9, below 10 and do subtraction.

Now onward after every step,  we get new divisor by doing addition of  Quotient & divisor

So, our new divisor is, 3(Quotient) + 3(Divisor) = 6

And bring the next group 34, down. Now our dividend = 134.

Step 3: Here apply the rule that, A new divisor can be multiplied by only that number which is suffixed to it. In this case,

1. If we suffix 1 to 6 it will be, 61 & 61 x 1 = 61
2. If we suffix 2 to 6 it will be, 62 & 62 x 2 = 124

Step 4: Now the new divisor is obtained by using rule, we get new divisor by doing addition of  new Quotient & divisor

Here, adding 2 to 62 equal to 64. So, the new divisor is 64 & remainder is 10.

At this stage we simply divide the 10 by 64 and the answer is, 0.156

The quotient already obtained is, 32. Hence the complete answer is, 32.156.

Ans, Square root of 10 34 = 32. 156.

e.g. vi) Find the square root of 384557

Step 1: Divide the given number in to groups of two digits,  starting from right. At the end if single digit is left, consider it as a one more group.

Here, 384557 = 38 45 57

Start from right, take last two digits 57, in one group,  next group is 45 & one more group is 38.

Step 2: We will start with first group i.e. 38, Here 36 is a  perfect square number just smaller than 38. Obtained by, 6 x 6.

Hence, we put 6 in divisor column & other 6 in quotient column. Write their product 36, below 38 and do subtraction.

Now onward after every step,  we get new divisor by doing addition of  Quotient & divisor

So, our new divisor is, 6(Quotient) + 6(Divisor) = 12

And bring the next group 45, down. Now our dividend = 245.

Step 3: Here apply the rule that, A new divisor can be multiplied by only that number which is suffixed to it. In this case,

1. If we suffix 1 to 12 it will be, 121 & 121 x 1 = 121
2. If we suffix 2 to 12 it will be, 122 & 122 x 2 = 244

Step 4:  Repeat the rule, we get new divisor by doing addition of  Quotient & divisor.

So our new divisor is equal to, 122 + 2 = 124

If we add suffix 1 to quotient 124 then 1241 will be  greater than 157. But the number of digit in quotient is equal to the number of groups in divisor. So, we add 0 as a suffix.

Now Remainder is 157 & Q is 1240, we simply divide it,

157/1240 = 0.126

We already have a quotient 620, So our final answer is 620.126

Ans, Square root of 384557 = 620.126