How to find Square Root of 400
Square of 400 :
- 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 20
- To find the square of 20,we multiply 20 by the number itself i.e. by 20 and we write it as follows (400)2 = 20*20= 400
Square root of 400:
- 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 20 can be written as,
√400 = √ (20*20) = 20
Where √ is the symbol which is called as radical sign.
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 400 is the positive perfect square which has two roots +20 and -20 also.
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √400 = √(-20)*(-20) = -20 and √400 = √(20)*(20) = 20
Similarly,
- (-20)*(-20) = (-20)2 = +400 and (+20)*(+20) = (+20)2 = 400
Methods to find square root of perfect square like 400:
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 400, first we subtract 1 from it. 400 – 1 =399
- Then next odd number is 3, so we have to subtract it from 399 399– 3 = 396
- 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 400 by repeated subtraction method as follows:
400-1= 399
399-3= 396
396-5= 391
391-7= 384
384-9= 375
375- 11=364
364-13= 351
351-15= 336
336-17= 319
319-19= 300
300-21= 279
279-23= 256
256-25= 231
231-27= 204
204-29= 175
175-31= 144
144-33= 111
111-35= 76
76- 37= 39
39-39= 0
Thus, here the total odd numbers used are 1, 3, 5, 7, 9, 11, 13, 15 ,17,19,21,23,25,27,29 ,31,33,35and 39which are 20 in numbers.
Hence, the square root of 400 by repeated subtraction method is 20.
-
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 400 by prime factorization method.
As 400 is even number hence it must be divisible by prime number 2 and 5
400÷2= 200
200÷2= 100
100÷2=50
50÷2=25
25÷5=5
5÷5=1
- Thus, the prime number 2 and 5 used to get remainder as 1 are 2 ,2,2,2,5,5
Thus, 400= 2*2*2*2*5*5= 2^2*2^2*5^2
And 400= (2*2*5)
- By taking square root on both sides, we get
√400 = √(2*2*5) = √(2*2) √(5*5)√(2*2)
√400 = (2*2*5)= 20
- Thus, we found the square root of 400 as 20 by using prime factorization method.
Multiple choice questions:
1) the correct expontenial form of square root 400 is—–
a) 2^2*5^2
b) 2^2*2^2*5^2
c) 2^2*2^2*2^2
d) 2^2*5^2*5^2
Ans: b)2^2*2^2*5^2
2) the prime factorization of 400 is —–
a) 2*2*2*2*5*5
b) 2*2*5
c) 2*2*2*2*2*5
d) 2*3*4
Ans: a)2*2*2*2*5*5
3) 400 is not perfect square
a) True
b) False
c) Both a and b
d) None
Ans: b) false