Square Root of 123

The square root of 123 is ±11.0905365, where 11.0905365 is the positive solution of the equation x² = 123.

 

Finding the square root is just the inverse of squaring a number and hence, squaring 11.0905365 will result in 123.

 

The square root of 123 is written as √123 in radical form, where the ‘√’  sign is called the “radical”  sign. In exponential form, it is written as (123)^1/2

We can find the square root of 123 through various methods. They are:
i) Prime factorization method
ii) Long division method
iii) Approximation/Estimation method
 

The prime factorization of 123 can be found by dividing 123 by prime numbers and continuing to divide the quotients until they can’t be separated anymore. After factorizing 123, make pairs out of the factors to get the square root. If there exist numbers that cannot be made pairs further, we place those numbers with a “radical” sign along with the acquired pairs

So, Prime factorization of 123 = 41 × 3   
But here in the case of 123, no pair of factors are obtained but a single 3 and a single 41 are remaining
So, it can be expressed as  √123 = √(41 × 3) = √123
√123 is the simplest radical form of √123
 

This method is used for obtaining the square root for non-perfect squares, mainly. It usually involves the division of the dividend by the divisor, getting a quotient and a remainder too, where the dividend is the number we are finding the square root of.

 

Follow the steps to calculate the square root of 123:

 

Step 1: Place the number 123 just the same as the image, starting from right to left, and
              draw a bar above the pair of digits.

 Step 2:  The first number under “bar” is 1
               Now, find the greatest number whose square is less than or equal to 1. Here, it is
               1, Because 1²=1.

Step 3:   now divide 1 by 1 (the number we got from Step 2) such that we get 1 as a 
             quotient, and we get a remainder.  Double the divisor 1, we get 2, and then the 
             largest possible number A1=1 is chosen such that when 1 is written beside the new
             divisor 2, a 2-digit number is formed →21, and multiplying 1 with 21 gives 21, which
            is less than 23.
             Repeat this process until you reach the remainder of 0. We are left with the
             the remainder, 11900 (refer to the picture), after some iterations and
              keeping the division till here, at this point

 Step 4: The quotient obtained is the square root. In this case, it is 11.090….
 

Follow the steps below:

Step 1: find the square roots of the perfect squares above and below 123
             Below : 121 → square root of 121 = 11 ….(i)
             Above : 144 →square root of 144 = 12  …..(ii)

Step 2: Dividing 123 with one of 11 or 12 
             If we choose 11 and divide 123 by 11, we are getting 11.18181  ….(iii)

Step 3:  find the average of 11 (from Step (i)) and 11.18181 (from Step (iii))
            (11+11.18181)/2 = 11.0909 

 

 Hence, 11.0909 is the approximate square root of 123
 

Exponential form

1) An algebraic expression that includes an exponent. It is a way of expressing the numbers raised to some power of their factors. It includes continuous multiplication involving base and exponent. Ex: 2 ⤬ 2 ⤬ 2 ⤬ 2 = 16 Or, 2 4 = 16, where 2 is the base, 4 is the exponent 

2) Factorization   
Expressing the given expression as a product of its factors Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3

3) Prime Numbers 
Numbers which are greater than 1, having only 2 factors as →1 and Itself.
Ex: 1,3,5,7,....

4) Rational numbers and Irrational numbers
The numbers that can be expressed as p/q, where p and q are integers and q is not equal to 0 are called Rational numbers. Numbers that cannot be expressed as p/q, where p and q are integers and q is not equal to 0 are called Irrational numbers. 

5) Perfect and non-perfect square numbers
Perfect square numbers are those numbers whose square roots do not include decimal places. Ex: 4,9,25, Non-perfect square numbers are those numbers whose square roots comprise decimal places. Ex :3, 8, 24.