How to find Square Root of 3249
Square of 57
- 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 57
- To find the square of 57 we multiply 57 by the number itself i.e. by 57 and we write it as follows (57)2 = 57*57= 3249
Square root of 3249
- 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 3249 can be written as,
√3249= √ (57*57) = 57
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 57 and square root of 3249 as
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 3249 is the positive perfect square which has two roots +57 and -57 also.
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √3249 = √(-57)*(-57) = 57 and √3249 = √(57)*(57) = 57
Similarly,
- (-57)*(-57) = (-57)2 = +3249 and (+57)*(+57) = (+57)2 = 3249
Methods to find square root of perfect square like 3249
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 3249 first we subtract 1 from it.
3249– 1 = 3248
- Then next odd number is 3, so we have to subtract it from 3248
3248– 3 = 3245
- 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 3249 by repeated subtraction method as follows:
3249-1=3248
3248-3=3245
3245-5 = 3240
3240- 7= 3233
3233- 9 =3224
3224-11=3213
3213-13= 3200
3200-15=3185
3185-17=3168
3168-19=3149
3149-21=3128
3128-23=3105
3105-25=3080
3080-27=3053
3053-29=3024
3024-31=2993
2993-33=2960
2960-35=2925
2925-37=2888
2888-39=2849
2849-41=2808
2808-43=2765
2765-45=2720
2720-47=2673
2673-49=2624
2624-51=2573
2573-53=2520
2520-55=2465
2465-57=2408
2408- 59 = 2349
2349- 61 =2288
2288- 63 = 2225
2225- 65 =2160
2160-67=2093
2093-69=2024
2024-71= 1953
1953-73=1880
1880-75=1805
1805-77=1728
1728-79=1649
1649-81=1568
1568-83=1485
1485-85=1400
1400-87=1313
1313-89=1224
1224-91=1133
1133-93=1040
1040-95=945
945-97=848
848-99=749
749-101=648
648-103=545
545-105=440
440-107=333
333-109=224
224-111=113
113-113=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 ,107,109,111,113which are 57in numbers.
- Hence, the square root of 3249 by repeated subtraction method is57
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 3249 by prime factorization method.
- As 3249 is odd number hence it must be divisible by prime number 3 ,19
- So, we start from prime number 3 here.
3249÷3=1083
1083÷3=361
361÷19=19
19÷19=1
- Thus, the prime number 3,19used to get remainder as 1 are 3,3,19,19
Thus, 3249=3*3*19*19
And 3249=3*3*19*19
- By taking square root on both sides, we get
√3249=√(3*3*19*19)=√(3*3)√(19*19)=3*19=57
- Thus, we found the square root of 3249 as 57 by using prime factorization method.
Multiple choice questions:
1) 3249=?
a) 3*3*19*19
b)3*3*3*19*19
c)3*3*3*3*19*19*19
d) none of these
Ans: a) 3*3*19*19
2) the value of number of odd numbers required to get 0 is the required——-
a) square root
b) cube root
c) fourth root
d) all of these
Ans: a) square root
3) (-57)*(-57) = (-57)2 = +3249
a) true
b) false
Ans: a) true