How to find Square Root of 3969
Square of 63
- In mathematics, finding of square of any number is mostly easy because when we multiply the same number with itself then we will get the square of that number.
For example:
- Let us suppose we have to find the square of any number say X, then we multiply X by itself i.e. X and we will get its square as Y. It can be written as (X)2 = X*X= Y
- In similar way we find the square of 63
- To find the square of 63 we multiply 63 by the number itself i.e. by 63 and we write it as follows (63)2 = 63*63= 3969
Square root of 3969
- Now, in reverse manner if we have to find the square root of Y. The square root of Y is that single value which when multiplied with itself gives the value Y.
- That means, √Y = √(X*X) = X
Where √ is the symbol named as radical.
For example:
- The square root of 3969 can be written as,
√3969= √ (63*63) = 63
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 63 and square root of 3969 as
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 3969 is the positive perfect square which has two roots +63 and -63 also
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √3969 = √(-63)*(-63) = 63 and √3969 = √(63)*(63) = 63
Similarly,
- (-63)*(-63) = (-63)2 = +3969 and (+63)*(+63) = (+63)2 = 3969
Methods to find square root of perfect square like 3969
There are many methods to find the square root of perfect squares out of which we see the following method in detail.
- Repeated Subtraction Method
- Prime factorization method
Repeated Subtraction Method:
- In repeated subtraction method, we have to subtract the consecutive odd numbers starting from 1, from the perfect square number whose square root we have to find.
- e. to find square root of 3969 first we subtract 1 from it.
3969– 1 = 3968
- Then next odd number is 3, so we have to subtract it from 3968
3968– 3 = 3965
- In this way, we subtract the consecutive odd numbers from the corresponding values obtained after subtraction continuously till we get final value as 0.
- And the value of number of odd numbers required to get 0 is the required square root.
For example:
- We find the square root of 3969 by repeated subtraction method as follows:
3969-1=3968
3968-3 = 3965
3965-5=3960
3960- 7= 3953
3953- 9 =3944
3944-11=3933
3933-13= 3920
3920-15=3905
3905-17=3888
3888-19=3869
3869-21=3848
3848-23=3825
3825-25=3800
3800-27=3773
3773-29=3744
3744-31=3713
3713-33=3680
3680-35=3645
3645-37=3608
3608-39=3569
3569-41=3528
3528-43=3485
3485-45=3440
3440-47=3393
3393-49=3344
3344-51=3293
3293-53=3240
3240-55=3185
3185-57=3128
3128- 59 = 3069
3069- 61 =3008
3008- 63 = 2945
2945- 65 =2880
2880-67=2813
2813-69=2744
2744-71= 2673
2673-73=2600
2600-75=2525
2525-77=2448
2448-79=2369
2369-81=2288
2288-83=2205
2205-85=2120
2120–87=2033
2033-89=1944
1944-91=1853
1853-93=1760
1760-95=1665
1665-97=1568
1568-99=1469
1469-101=1368
1368-103=1265
1265-105=1160
1160-107=1053
1053-109=944
944-111=833
833-113=720
720-115=605
605-117=488
488-119=369
369-121= 248
248-123=125
125-125=0
- Thus, here the total odd numbers used are 1, 3, 5, 7, 9, 12, 13, 15, 17, 19, 21, 23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55 ,57 ,59 61,63,65,67,69 ,71,73,75,77,79,81 ,83,85 ,87,89,91,93,95 97,99,101,103,105 ,107,109,111,113,115,117,119,121,123 ,125which are 63 in numbers.
- Hence, the square root of 3969 by repeated subtraction method is 63
Prime Factorization method:
- In prime factorization method, we have to divide the perfect square number whose square root we have to find by prime number starting from 2, 3, 5… and so on till we get the remainder as 1.
- Initially we have to divide by prime number 2, if that number is not divisible by 2 then we have to take next prime number i.e. 3 and the process will be continued till we get remainder as 1.
- Finally, we have to make pairs of the prime numbers taken in the form of multiplication and then we have to take its square root.
For example:
- Following is the process to find the square root of 3969 by prime factorization method.
- As 3969 is odd number hence it must be divisible by prime number 3,7
- So, we start from prime number 3 here.
3960÷3=1323
1323÷3=441
441÷3=147
147÷3=49
49÷7=7
7÷7=1
- Thus, the prime number used to get remainder as 1 are 3,3,3,3,7,7
Thus, 3969=3*3*3*3*7*7
And 3969=3*3*3*3*7*7
- By taking square root on both sides, we get
√3969=√(3*3)(3*3)(7*7)=√(3*3)√(3*3)√(7*7)=3*3*7=63
- Thus, we found the square root of 3969 as 63 by using prime factorization method.
Multiple choice questions:
1) 3969 is a perfect square of ——–.
a)3*3*7
b)3*3*3*3*7*7
c)4*4*3*3
d) both b and c
Ans: a) 3*3*7
2) 3969= ?
a) 3*3*3*3*7*7
b)4*4*4*7*7*8
c)3*3*3*7*7*9
d) none of these
Ans: a) 3*3*3*3*7*7
3) 3969 is not a prime number.
a) true
b) false
Ans: a) true