How to find Square Root of 4225
Square of 65
- 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 65
- To find the square of 65 we multiply 65 by the number itself i.e. by 65 and we write it as follows (65)2 = 65*65= 4225
Square root of 4225
- 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 4225 can be written as,
√4225= √ (65*65) = 65
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 65 and square root of 4225 as
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 4225 is the positive perfect square which has two roots +65 and -65 also
- But, the positie square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √4225 = √(-65)*(-65) = 65 and √4225 = √(65)*(65) = 65
Similarly,
- (-65)*(-65) = (-65)2 = +4225 and (+65)*(+65) = (+65)2 = 4225
Methods to find square root of perfect square like 4225
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 4225 first we subtract 1 from it.
4225– 1 = 4224
- Then next odd number is 3, so we have to subtract it from 4224
4224– 3 = 4221
- 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 4225 by repeated subtraction method as follows:
4225-1=4224
4224-3 = 4221
4221-5=4216
4216- 7= 4209
4209- 9 =4200
4200-11=4189
4189-13= 4176
4176-15=4161
4161-17=4144
4144-19=4125
4125-21=4104
4104-23=4081
4081-25=4056
4056-27=4029
4029-29=4000
4000-31=3969
3969-33=3936
3936-35=3901
3901-37=3864
3864-39=3825
3825-41=3784
3784-43=3741
3741-45=3697
3696-47=3649
3649-49=3600
3600-51=3549
3549-53=3496
3496-55=3441
3441-57=3384
3384- 59 = 3325
3325- 61 =3264
3264- 63 = 3201
3201- 65 =3136
3136-67=3069
3069-69=3000
3000-71= 2929
2929-73=2856
2856-75=2781
2781-77=2704
2704-79=2625
2625-81=2544
2544-83=2461
2461-85=2376
2376–87=2289
2289-89=2200
2200-91=2109
2109-93=2016
2016-95=1921
1921-97=1824
1824-99=1725
1725-101=1624
1624-103=1521
1521-105=1416
1416-107=1309
1309-109=1200
1200-111=1089
1089-113=976
976-115=861
861-117=744
744-119=625
625-121= 504
504-123=381
381-125=256
256-127=129
129-129=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 ,125,127,129which are 65 in numbers.
- Hence, the square root of 4225by repeated subtraction method is 65
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 4225 by prime factorization method.
- As 4225 is odd number hence it must be divisible by prime number 5,13
- So, we start from prime number 5 here.
4225÷5=845
845÷5=169
169÷13=13
13÷13=1
- Thus, the prime number 5,13used to get remainder as 1 are 5,5,13,13
Thus, 4225=13*13*5*5
And 4225=13*13*5*5
- By taking square root on both sides, we get
√4225=√(5*5)(13*13)=√(5*5)√(13*13)=5*13=65
- Thus, we found the square root of 4225 as 65 by using prime factorization method.
Multiple Choice Questions:
1) Square root of 4225 by repeated subtraction method is ——-
a) 66
b) 65
c) 56
d) 67
Ans: b) 65
2) 4225=13*13—-*—-
a)5,5
b)6,6
c)7,7
d)9,9
Ans: a) 5,5
3) 4225 is divisible by prime no.——
a) 3
b)2
c)5
d)7
Ans: c)5