Square Root

Square root definition can be defined as a number which when multiplied by itself gives a as the product, then it is known as a square root. For example, since 


5 × 5 = 25, so \[\sqrt{25}\] = 5


All real numbers have two square roots, one is a positive square root and another one is a  negative square root. The positive square root is also referred to as the principal square root. Two positive numbers when multiplied results in a positive number because of having the same sign. As far as negative numbers are concerned, they don't have real square roots because a square is either positive or 0. Square root symbol is represented as  \[\sqrt{}\]

Perfect Squares

When two equal numbers are multiplied, it results in a perfect square. For example, 55=25

This is a basic property of the square root

Square Root Table

No

Value



No

Value



No

Value

1

1.000



21

4.583



41

6.403

2

1.414



22

4.690



42

6.481

3

1.732



23

4.796



43

6.557

4

2.000



24

4.899



44

6.633

5

2.236



25

5.000



45

6.708

6

2.449



26

5.099



46

6.782

7

2.646



27

5.196



47

6.856

8

2.828



28

5.292



48

6.928

9

3.000



29

5.385



49

7.000

10

3.162



30

5.477



50

7.071

11

3.317



31

5.568



51

7.141

12

3.464



32

5.657



52

7.211

13

3.606



33

5.745



53

7.280

14

3.742



34

5.831



54

7.348

15

3.873



35

5.916



55

7.416

16

4.000



36

6.000



56

7.483

17

4.123



37

6.083



57

7.550

18

4.243



38

6.164



58

7.616

19

4.359



39

6.245



59

7.681

20

4.472



40

6.325



60

7.746



61

7.810



81

9.000

62

7.874



82

9.055

63

7.937



83

9.110

64

8.000



84

9.165

65

8.062



85

9.220

66

8.124



86

9.274

67

8.185



87

9.327

68

8.246



88

9.381

69

8.307



89

9.434

70

8.367



90

9.487

71

8.426



91

9.539

72

8.485



92

9.592

73

8.544



93

9.644

74

8.602



94

9.695

75

8.660



95

9.747

76

8.718



96

9.798

77

8.775



97

9.849

78

8.832



98

9.899

79

8.888



99

9.950

80

8.944



100

10.000


Referring to this square root table, half of the energy will be saved. 

How To Find Square Root?

Now, since we know what a square root is, we can quickly jump into knowing how to find a square root of a number. There is no square root formula as such but well, there are two ways to find the square root of a number. They are the Prime Factorization Method and the Division Method. You will know how to solve the square root equation using these two methods.


Finding Square Root Using Prime Factorization Method 

To find the square root of a perfect square we have to follow the following steps:

Step 1) First resolve the given number into prime factors.

Step 2) Make pairs of similar factors.

Step 3) The product of prime factors, chosen one out of every pair, gives the square root of the given number. 

For example, find the square root of  24336.


Solution) Resolving 24336 into prime factors, we get: 


2

24336

2

12168

2

6084

2

3042

3

1521

3

507

13

169

13

13


1



24336 = \[\underline{2}\] × \[\underline{2}\] × \[\underline{2}\] ×  \[\underline{2}\] × \[\underline{3}\] ×  \[\underline{3}\]  × \[\underline{13}\]  × \[\underline{13}\]


\[\sqrt{24336}\] = 2 × 2 × 3 × 13 = 156


Finding Square Root Using Division Method

In case of large numbers; we find the square root by using the following steps: 

Step 1) Mark off the digits in pairs starting with the unit digit. Every pair and remaining one digit (if any) is called a period. 


Step 2) Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.


Step 3) In this step subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. This becomes the new dividend. 


Step 4) now, a new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of new divisor and this digit is equal to or just less than the new dividend.  

Solved Examples


Question 1) By using the table of square roots, find the values of 


i) \[\sqrt{13}\]

ii) \[\sqrt{83}\]

iii) \[\sqrt{150}\]

iv) \[\sqrt{36}\]


Solution 1)

i) Let x = 13

From the table, we find that when  x = 13, then \[\sqrt{x}\] 3.606.

Therefore, \[\sqrt{13}\] = 3.606.

ii) Let x = 83

From the table, we find that when  x = 83, then \[\sqrt{x}\] = 9.110.

Therefore,  \[\sqrt{83}\]=9.110.

iii) \[\sqrt{150}\] =  \[\sqrt{5 × 5 × 6}\] = 5 × \[\sqrt{6}\] = 5 × 2.449

Using the table for \[\sqrt{6}\],  = 12.245

Therefore, \[\sqrt{150}\] =  12.245

iv)\[\sqrt{368}\] = \[\sqrt{4 × 4 × 23}\] = 4 × \[\sqrt{23}\] = 4 × 4.796

Using the table for \[\sqrt{23}\], = 19.184

Therefore, \[\sqrt{368}\] = 19.184


Question 2) find the square root of  1764

 

Solution 2)

Resolving 1764 into prime factors, we get:


2

1764

2

882

3

441

3

147

7

49

7

7


1



1764 = \[\underline{2}\] × \[\underline{2}\] × \[\underline{3}\] × \[\underline{3}\] × \[\underline{7}\] × \[\underline{7}\]


\[\sqrt{1764}\] = 2 × 3 × 7 = 42


Question 3) Find the smallest number which when multiplied by 720 gives a perfect square number. 

i) Give the perfect square number so obtained?

ii) Find the square root of this perfect square number.


Solution 3) Resolving 720 into prime factors, we get:

720 = 2 × 2  ×  2  ×  2  ×  3  ×  3  ×  5.

Thus, 2,2,3 exist in pairs while 5 is alone.

So, we should multiply the given number by 5 to get a perfect square number.

i) Perfect square number so obtained = 720 × 8 = 3600

ii) Now, 3600 = 2  ×  2  ×  2  ×  2  ×  3  ×  3  ×  5 ×  5

Therefore, \[\sqrt{3600}\] =   2 × 2  ×  3  ×  5 = 60

FAQ (Frequently Asked Questions)

1. What is the relation between square numbers and square root?

When a square root is multiplied, it produces a square number. Thus, a square root has an inverse operation than the square number. Let’s take an example, the square root of 36 is 6 because 6 x 6 = 36. Finding the square root of a number can be much trickier than calculating the square number. Therefore, the modern calculator has the square root button to make the calculation easier.


2. What are the applications of square roots?

Square root can be very important in mathematical topics such as quadratic equations, algebra, geometry, and calculus.

3. How to find the square root of a number?

There are two methods to find square root i.e prime factorization method and division method. We can also learn how to find square roots without a calculator as nowadays many calculators have a square root function.