How to find Square Root of 144
Square of 12:
- 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 12.
- To find the square of 12, we multiply 12 by the number itself i.e. by12 and we write it as follows.
- (12)2 = 12*12 = 144
Square root of 144:
- 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 144 can be written as,
√144 = √ (12*12) = 12
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 12 and square root of 144 as
Note:-
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:-
- 144 is the positive perfect square which has two roots +12 and -12 also.
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √144 = √(-12)*(-12) = -12and √144 = √(12)*(12) = 12
Similarly,
- (-12)*(-12) = (-12)2 = +144 and (+12)*(+12) = (+12)2 = 144
Methods to find square root of perfect square like 144:
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 144, first we subtract 1 from it. 144 – 1 = 143
- Then next odd number is 3, so we have to subtract it from 143. 143 – 3 = 140
- 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 144 by repeated subtraction method as follows:
144 – 1 = 143
143 – 3 = 140
140 – 5 = 135
135 – 7 = 128
128 – 9 = 119
119 – 11 = 108
108– 13 = 95
95– 15 = 80
80 – 17 = 63
63– 19 = 44
44 – 21 = 23
23 – 23 = 0
- Thus, here the total odd numbers used are 1, 3, 5, 7, 9, 12, 13, 15, 17, 19, 21, 23 and which are 12 in numbers.
- Hence, the square root of 144 by repeated subtraction method is 12.
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 144 by prime factorization method.
- As 144 is even number hence it must be divisible by prime number 2.
- So, we start from prime number 2 here.
144 ÷ 2 = 72
72÷ 2 = 36
36 ÷ 2 = 18
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
Thus, the prime number 2, 3 used to get remainder as 1 are 2, 2, 2, 2, 3, 3.
Thus, 144 = 2*2*2*2*3*3
And 144 = 2*2*2*2*3*3
By taking square root on both sides, we get
√144 = √(2*2*2*2*3*3) = √2*2 √2*2 = √3*3
√144 = 2*2*3 = 12
Thus, we found the square root of 144 as 12 by using prime factorization method.
Multiple Choice Questions:-
1) In repeated subtraction method, the consecutive numbers subtracted from the number whose square root we have to find are
a) Odd numbers
b) Even numbers
c) Prime numbers
d) Successive odd numbers
Ans: d) successive odd numbers
2) 144 is the
a) Odd number
b) Even number
c) Non -Perfect square
d) Both b and c
Ans: a) odd number
3) The square root of 144 by prime factorization method is
a) -144
b) 12
c) -12
d) 144
Ans: b) 12