How to find Square Root of 9
Square of 3:
- 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 3.
- To find the square of 3, we multiply 3 by the number itself i.e. by3 and we write it as follows (3)2 = 3*3 = 9
Square root of 9:
- 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 9 can be written as,
√9 = √ (3*3) = 3
Where √ is the symbol which is called as radical sign.
- In short, we remember square of 3 and square root of 9 as –
Note:
- Every positive real number has two roots.
- The square of any negative number is always the positive number.
For example:
- 9 is the positive perfect square which has two roots +3 and -3 also.
- But, the positive square root value is taken mostly which is called as principal square root or non-negative square root.
- Hence, √9 = √(-3)*(-3) = -3and √9 = √(3)*(3) = 3
Similarly,
- (-3)*(-3) = (-3)2 = +9 and (+3)*(+3) = (+3)2 = 9
Methods to find square root of perfect square like 9:
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 9, first we subtract 1 from it. 9 – 1 = 8
- Then next odd number is 3, so we have to subtract it from 8. 8 – 3 = 5
- 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 9 by repeated subtraction method as follows:
9 – 1 = 8
8 – 3 = 5
5 – 5 = 0
Thus, here the total odd numbers used are 1, 3, 5 and which are 3 in numbers.
- Hence, the square root of 9 by repeated subtraction method is 3.
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 9 by prime factorization method.
- As 9 is odd number hence it must be divisible by prime number 3.
- So, we start from prime number 3 here.
9 ÷ 3 = 3
3 ÷ 3 = 1
Thus, the prime number 3 used to get remainder as 1 are 3, 3.
Thus, 9 = 3*3= 32
And 9 = (3*3)
By taking square root on both sides, we get
√9 = √(3*3)
√9 = 3
Thus, we found the square root of 9 as 3 by using prime factorization method.
Multiple Choice Questions:
1) In repeated subtraction method, the consecutive numbers subtracted from the 9 whose square root we have to find are\
a) Odd numbers 1, 3, 5
b) Even numbers
c) Prime numbers
d) Successive odd numbers which are 1, 3, 5
Ans: d) successive odd numbers which are 1, 3, 5
2) The square root of 9 by prime factorization method is
a) 4
b) 3
c) 9
d) 6
Ans: b) 3
3) Squares of +3 and -3 are
a) +9 and -9 respectively
b) +9 and +9 respectively
c) -9 and +9 respectively
d) None
Ans: b) +9 and +9 respectively