How to find Square Root of 4096
Square of 64
- 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 64
- To find the square of 64 we multiply 64 by the number itself i.e. by 64 and we write it as follows (64)2 = 64*64= 4096
Square root of 4096
- 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 4096 can be written as,
√4096= √ (64*64) = 64
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 64 and square root of 4096 as
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 4096 is the positive perfect square which has two roots +64 and -64 also
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √4096 = √(-64)*(-64) = 64 and √4096 = √(64)*(64) = 64
Similarly,
- (-64)*(-64) = (-64)2 = +4096 and (+64)*(+64) = (+64)2 = 4096
Methods to find square root of perfect square like 4096
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 4096 first we subtract 1 from it.
4096– 1 = 4095
- Then next odd number is 3, so we have to subtract it from 4095
4095– 3 = 4092
- 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 4096 by repeated subtraction method as follows:
4096-1=4095
4095-3 = 4092
4092-5=4087
4087- 7= 4080
4080- 9 =4071
4071-11=4060
4060-13= 4047
4047-15=4032
4032-17=4015
4015-19=3996
3996-21=3975
3975-23=3952
3952-25=3927
3927-27=3900
3900-29=3871
3871-31=3840
3840-33=3807
3807-35=3772
3772-37=3735
3735-39=3696
3696-41=3655
3655-43=3612
3612-45=3567
3567-47=3520
3520-49=3471
3471-51=3420
3420-53=3367
3367-55=3312
3312-57=3255
3255- 59 = 3196
3196- 61 =3135
3135- 63 = 3072
3072- 65 =3007
3007-67=2940
2940-69=2871
2871-71= 2800
2800-73=2727
2727-75=2652
2652-77=2575
2575-79=2496
2496-81=2415
2415-83=2332
2332-85=2247
2247–87=2160
2160-89=2071
2071-91=1980
1980-93=1887
1887-95=1792
1792-97=1695
1695-99=1596
1596-101=1495
1495-103=1392
1392-105=1287
1287-107=1180
1180-109=1071
1071-111=960
960-113=847
847-115=732
732-117=615
615-119=496
496-121= 375
375-123=252
252-125=127
127-127=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 ,125,127 which are 64 in numbers.
- Hence, the square root of 4096 by repeated subtraction method is 64
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 4096 by prime factorization method.
- As 4096 is even number hence it must be divisible by prime number 2
- So, we start from prime number 2 here.
4096÷2=2048
2048÷2=1024
1024÷2=512
512÷2=256
256÷2=128
128÷2=64
64÷2=32
32÷2=16
16÷2=8
8÷2=4
4÷2=2
2÷2=1
- Thus, the prime number 2 used to get remainder as 1 are 2,2,2,2,2,2,2,2,2,2,2,2
Thus, 4096=2*2*2*2*2*2*2*2*2*2*2*2
And 4096=2*2*2*2*2*2*2*2*2*2*2*2
- By taking square root on both sides, we get
√4096=√(2*2)(2*2)(2*2)(2*2)(2*2
(2*2)=√(2*2)√(2*2)√(2*2)√(2*2)√(2*2)√(2*2)=2*2*2*2*2*2=64
- Thus, we found the square root of 4096 as 64 by using prime factorization method.
Multiple choice questions:
1) the square root of ————-by repeated subtraction method is 64
a) 4095
b)4096
c)4084
d)4888
Ans: b)4096
2) 4096 is the positive perfect square which has——– roots
a) 2
b)4
c)7
d)3
Ans: a) 2
3) Every positive real number has—— roots.
a) 2
b)9
c)2
d)3
Ans: c)2