**Perfect squares:**

First we will see about finding the square root of perfect square with examples-

Let we have to find the square root of the square **82****8****1**

1^{s}^{t} of all I want you to remember the below table-

The last digit of the perfect square | The last digit of the square root |

1 | 1 or 9 |

4 | 2 or 8 |

9 | 3 or 7 |

6 | 4 or 6 |

5 | 5 |

0 | 0 |

Note:A perfect square will never end with digit “2,3,7 or 8”

- So, now in our example the number end up with a digit
**1**so the number**8281**is perfect square.

Now follow the below step- - 1 st of all since the number ends with digit 1, square root will end with digit 1 or 9. So the digit of

extreme right of the number is either 1 or 9. - Now take the number 8281 and find that the number lies between 8100(perfect square of 90) and

10000(perfect square of 100). At this stage the answer should lie between 90 and 100. - Now find the numbers ending with digit 1 or 9 between 90 and 100. Then we find two numbers

91 and 99. So answer is either 91 or 99. - Now find is 8281 is closer to smaller number 8100 or bigger number 10000. If 8281 is closer two

smaller numbers then the answer is 91, because 91 is the smaller then 99. And if 8281 is closer to

bigger number 10000 then answer is the bigger number 99. But obviously 8281 is closer to 8100,

so the answer is 91.

So the final answer is **91.**

If the given number is 9801 then it is closer to 10000. Then the answer will be 99.

**General square:**

With this method square root of any general square can be evaluated. Specially it is easy method to find

square root of 5-digit and higher digit number, since the above discussed technique is only deal with

perfect square up to 4-digit number conveniently.

There is two points to be remembered in this method:-

- After every step, add quotient to the divisor and get a new divisor.
- New divisor can be multiplied by only that number which is suffixed to the new divisor

### Examples

**Question 1**

**Question 1**

*Find the square root of 1156*

Now1^{s}^{t} of all, group the number of a pair of two digits from the right side then grouping will be like as

below __11 56__

Now divide the number keep in mind the above two points,

Now detailed step of the above example:-

- 1 st try to find a perfect square number less than 11. So, 9 is the perfect square less than 11. 9 is the

square of 3. So divide eleven by 3 and put 3 in quotient also. Then we reminder as 2. - In 2 nd step add the divisor 3 and quotient 3 to get a new divisor which is 6(3+3). Now bring down

the other pair 56, now our new dividend is 256. Now according to 2 nd remember point new divisor

can be multiplied only with that number which is suffixed to it, so in this case (64×4=256),our

new divisor is 64 and reminder is 0. - This process will continue until the reminder is zero.
- And also the answer is the quotient after obtaining the reminder zero.

So the square root of the 1156 is 34. (Anybody can check the answer with calculator).

Now take another example of a square number of 7 digits.

__Q____ues____t____io____n 2 __

__Q____ues____t____io____n 2__*Find square root of 1476225.*

** Answer: **– Various step for finding the square roots are given below,

1^{s}^{t} of all mark off the digits in pair from the right side or from the unit‟s digit. The we get the given number as __1____ ____47____ ____62____ ____2____5____ __in pairs. Now find perfect square number less than or equal to 1. We find 1 is perfect square of 1. So put 1 in divisor as well as quotient.

- Now bring down the next pair 47. And put new divisor (as 1 st remember point) 2(1+1). Then put2 again in the suffix of the new divisor as well as in the quotient (according to 2 nd remember

point). Then put 44(22×2) below 47, and reminder is 3. - Now bring down the next pair of 62 and new dividend is 362. Now put 24 as new divisor (as 1 st remember point). And put 1 suffixed to 24 in new divisor as well as in the quotient. So new

divisor is 241. Put 241 below 362 as (241×1=241). So reminder is 121. - Now bring down the next pair 25 and new dividend is 12125. New divisor is 242(241+1). Now

put 5 as quotient and 2425 as new divisor. Because 2425×5=12125. So reminder is zero. And

final quotient we get is 1215.

$$\sqrt{1476225}=1215$$

You can check the answer using calculator. Now illustrations is given below,

__Q____ues____t____io____n 3__

__Q____ues____t____io____n 3__*Evaluate square root of \(\sqrt{0.9}\) up to 3 decimal places.*

Answer Illustration is given below,

Pairing is done after the decimal point 0.90 00 00Since we need the square root up to 3 decimal place,so we need to have 6 digit after decimal point.That ‘s why we put 5 zeros after 9 to make 6 digits after decimal point

So

**\(\sqrt{0.9}=.948 (3 decimals)\)**