We have divided the study of square roots in two parts. In this chapter we will study how to find the square roots of perfect squares; in the ‘Advanced Level’ we will study the square roots of general numbers. Most school and college exams ask the square roots of perfect squares. Therefore, this chapter is very useful to students giving such exams. Students of higher classes and other researchers will find the Chapter 13 useful as they will be able to study the Vedic Mathematics approach to calculate square roots of any given number—perfect as well as imperfect.
The need to find perfect square roots arises in solving linear equations, quadratic equations and factorizing equations. Solving square roots is also useful in geometry while dealing with the area, perimeter, etc. of geometric figures. The concepts of this chapter will also be useful in dealing with the applications of the Theorem of Pythagoras.
The technique of finding square roots of perfect squares is similar to the technique of finding the cube root of perfect
cubes. However, the former has an additional step and hence it is discussed after having dealt with cube roots.
WHAT IS SQUARE ROOT
To understand square roots it will be important to understand what are squares. Squaring of a number can be defined as multiplying a number by itself. Thus, when we multiply 4 by 4 we are said to have ‘squared’ the number 4.
The symbol of square is represented by putting a small 2 above the number.
E.g. (a) 42 = 4 * 4 = 16
(b) 52 = 5 * 5 = 25
From the above example we can say that 16 is the square of 4, and 4 is the ‘square root’ of 16. Similarly, 25 is the square of 5, and 5 is the square root of 25.
METHOD
To find the square roots it is necessary to be well-versed with the squares of the numbers from 1 to 10. The squares are given below. Memorize them before proceeding.
| NUMBER | SQUARE |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
In the chapter dealing with perfect cube roots we observed that if the last digit of the cube is 1 the last digit of the cube
root is also 1. If the last digit of the cube is 2 then the last digit of the cube root is 8 and so on. Thus, for every number there was a unique corresponding number.
However in square roots we have more than one possibility for every number. Look at the first row. Here, we have 1 in the number column and 1 in the square column. Similarly, in the ninth row we have 9 in the number column and 1 (of 81) in the square column. Thus, if the number ends in 1, the square root ends in 1 or 9 (because 1 * 1 is one and 9 * 9 is eighty-one). Do not worry if you do not follow this immediately. You may glance at the table below as you read these explanations, then all will be clear.
•Similar to the 1 and 9 relationship, if a number ends in 4 the square root ends in 2 or 8 (because 2 * 2 is four and 8 * 8 is sixty-four)
•If a number ends in 9, the square root ends in 3 or 7 (because 3 * 3 is nine and 7 * 7 is forty-nine)
•If a number ends in 6, the square root ends in 4 or 6 (because 4 * 4 is 16 and 6 * 6 is 36)
•If the number ends in 5, the square root ends in 5 (because 5 * 5 is twenty-five)
•If the number ends in 0, the square root also ends in 0 (because 10 * 10 is 100)
On the basis of such observations, we can form a table as given below:
| The Last Digit of the 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 |
Whenever we come across a square whose last digit is 9, we can conclude that the last digit of the square root will be 3 or 7. Similarly, whenever we come across a square whose last digit is 6, we can conclude that the last digit of the square root will be 4 or 6 and so on…
Now, I want you to look at the column on the left. It reads ‘Last digit of the square’ and the numbers contained in the column are 1,4, 9, 6, 5 and 0. Note that the numbers 2, 3, 7 and 8 are absent in the column. That means there is no perfect square which ends with the numbers 2, 3, 7 or 8. Thus we can deduct a rule:
‘A perfect square will never end with the digits 2, 3, 7 or 8’
At this point we have well understood how to find the last digit of a square root. However, in many cases we will have two possibilities out of which one is correct. Further, we do not know how to find the remaining digits of the square root. So we will solve a few examples and observe the technique used to find the complete square root.
Before proceeding with the examples, I have given below a list of the squares of numbers which are multiples of 10 up to 100. This table will help us to easily determine the square roots.
| NUMBER | SQUARE |
| 10 | 100 |
| 20 | 400 |
| 30 | 900 |
| 40 | 1600 |
| 50 | 2500 |
| 60 | 3600 |
| 70 | 4900 |
| 80 | 6400 |
| 90 | 8100 |
| 100 | 10000 |
(Q) Find the square root of 7744.
•The number 7744 ends with 4. Therefore the square root ends with 2 or 8. The answer at this stage is __2 or 8.
•Next, we take the complete number 7744. We find that the number 7744 lies between 6400 (which is the square of 80) and 8100 (which is the square of 90).
70 – 4900
80 – 6400
>
7744
90 – 8100
100 – 10000
The number 7744 lies between 6400 and 8100. Therefore, the square root of 7744 lies between the numbers 80 and 90.
•From the first step we know that the square root ends with 2 or 8. From the second step we know that the square root lies between 80 and 90. Of all the numbers between 80 and 90 (81, 82, 83, 84, 85, 86, 87, 88, 89) the only numbers ending with 2 or 8 are 82 or 88. Thus, out of 82 or 88, one is the correct answer.
(Answer at this stage is 82 or 88).
•Observe the number 7744 as given below:
80 – 6400
>
7744
90 – 8100
Is it closer to the smaller number 6400 or closer to the bigger number 8100?
If the number 7744 is closer to the smaller number 6400 then take the smaller number 82 as the answer. However, if it is closer to the bigger number 8100, then take 88 as the answer.
In this case, we observe that 7744 is closer to the bigger number 8100 and hence we take 88 as the answer.
The square root of 7744 is 88.
(Q) Find the square root of 9801.
•The last digit of the number 9801 is 1 and therefore the last digit of the square root will be either 1 or 9. The answer at this stage is 1 or 9.
•Next, we observe that the number 9801 lies between 8100 (which is the square of 90) and 10000 (which is the square of 100). Thus, our answer lies between 90 and 100. Our possibilities at this stage are:
91, 92, 93, 94, 95, 96, 97, 98, 99
•However, from the first step we know that the number ends with a 1 or 9. So, we can eliminate the numbers that do not end with a 1 or 9.
91, 92, 93, 94, 95, 96, 97, 98, 99
•The two possibilities at this stage are 91 or 99. Lastly, we know that the number 9801 is closer to the bigger number 10000 and so we take the bigger number 99 as the answer.
90 – 8100
>
9801
100 – 10000
(Q) Find the square root of 5184.
•5184 ends in 4. So the square root ends in either 2 or 8 (Answer = 2 or 8)
•5184 is between 4900 and 6400. So the square root is between 70 and 80. Combining the first two steps, the only two possibilities are 72 and 78
•Out of 4900 and 6400, our number 5184 is closer to the smaller number 4900 (70 * 70). Thus, we take the smaller number 72 as the correct answer
(Q) Find the square root of 2304.
•2304 ends in 4 and so the root either ends in a 2 or in a 8
•2304 lies between 1600 and 2500. So, the root lies between 40 and 50.
•Thus, the two possibilities are 42 and 48.
•Lastly, the number 2304 is closer to the bigger number 2500. Hence, out of 42 and 48 we take the bigger number 48 as the correct answer.
(Q) Find the square root of 529.
•529 ends with a 9. The answer is 3 or 7.
•It lies between 20 and 30. The possibilities are 23 or 27.
•529 is closer to the smaller number 400 and hence 23 is the answer.
We have seen five different examples and calculated their square roots. However, the final answer in each case is always a two-digit number. In most exams and even in general life, one will come across squares whose roots are a two-digit answer. Thus, the above examples are sufficient and there is no necessity to stretch the concept further. However, we will study a couple of examples involving big numbers so that you will understand the fundamentals thoroughly.
(Q) Find the square root of 12544.
The number 12544 ends with a 4. So, the square root ends with 2 or 8. The answer at this stage is 2 or 8.
•Further, we know that the square of 11 is 121 and so the square of 110 is 12100. Similarly, the square of 12 is 144 and so the square of 120 is 14400.
90 – 8100
100 – 10000
110 – 12100
>
12544
120 – 14400
•The number 12544 lies between 12100 (which is the square of 110) and 14400 (which is the square of 120). Thus, the square root of 12544 lies between 110 and 120.
•But we know that the square root ends with 2 or 8. Hence, our only possibilities are 112 or 118.
•Lastly, 12544 is closer to the smaller number 110 and hence we take the smaller possibility 112 as the answer. The square root of 12544 is 112.
(Q) Find the square root of 25281.
•The number 25281 ends with a 1. Therefore the square root ends with a 1 or a
9.The answer at this stage is 1 or 9.
•We know that the square of 15 is 225 and therefore the square of 150 is 22500. Similarly, the square of 16 is 256 and therefore the square of 160 is 25600.
150 – 22500
>
25281
160 – 25600
•We know that the root lies between 150 and 160 and hence the only possibilities are 151 and 159.
•Lastly, the number 25281 is closer to the bigger number 25600 and hence we take the bigger number 159 as the correct answer.
We have thus seen that the concept can be expanded to numbers of any length.
COMPARISON
As usual, we will be comparing the normal technique of calculation with our approach. In the traditional method of calculating square roots we use prime numbers as divisors.
Prime numbers are numbers which can be divided by themselves and by 1 only. They will not come in the multiplication table of any other number. They include numbers like 2, 3, 5, 7, 11, 13 and so on.
Let us say you want to find the square root of 256. Then, the process of calculating the square root of 256 is as explained below.
| 2 | 256 |
| 2 | 128 |
| 2 | 64 |
| 2 | 32 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
First we divide the given number 256 by the prime number 2 and get the answer as 128.
•Next, we divide 128 by 2 and get the answer 64
•64 divided by 2 gives 32
•32 divided by 2 gives 16
•16 divided by 2 gives 8
•8 divided by 2 gives 4
•4 divided by 2 gives 2
•2 divided by 2 gives 1
(We terminate the division when we obtain 1).
Thus, 256 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2.
To obtain the square root, for every two similar numbers, we take one number. So, we form four pairs containing two 2’s each. From every pair we take one 2.
It can be represented as:
256 = 2 * 2 * 2 * 2 * 2 * 2 * 2 8 2
2 * 2 * 2 * 2
Thus, the square root of 256 is 16.
(Q) Find the square root of 196 using prime factors.
| 2 | 196 |
| 2 | 98 |
| 7 | 49 |
| 7 | 7 |
| 1 |
196 = 2 * 2 * 7 * 7
2 * 7 = 14
Therefore, the square root of 196 is 14.
From the above two examples it is clear that the prime factor method of calculating square roots is time-consuming and tedious. Further, if small numbers like 256 and 196 take such a lot of time, one can imagine how difficult it will be to calculate the square roots of numbers like 8281, 7744, etc. Some people find it simply impossible to calculate the square roots of such numbers using the prime factor technique. Hence, the alternative approach as mentioned in this chapter will be of immense utility to the student.
(Q) Calculate the square root of 576.
Prime Factor Method Current Method
| 2 | 576 |
| 2 | 288 |
| 2 | 144 |
| 2 | 72 |
| 2 | 36 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
20=400
24
576><26
26
30 = 900
= 24
From the comparison we can see that the method described in this chapter is much faster and the chances of making a mistake are greatly reduced. Further, while the prime factor method will prove extremely tedious to calculate the square roots of numbers like 4356 and 6561, the current method will help us calculate them instantly!
EXERCISE
PART A
Q. (1) Find the square roots of the following numbers with the aid of writing material.
(1) 9216
(2) 7569
(3) 5329
(4) 3364
(5) 1681
PART B
Q. (1) Find the square roots of the following numbers without the aid of writing material.
(1) 9801
(2) 5625
(3) 1936
(4) 3481
(5) 1369
PART C
Q. (1) Find the square roots of the following numbers with or without the aid of writing material.
(1) 12769
(2) 15625
(3) 23104
(4) 11881
