Square Root of 123
2026-02-28 00:47 Diff
https://brightchamps.com/en-us/math/algebra/square-root-of-123

40251 Learners

Last updated on January 16, 2026

The square root of 123 is a value “y” such that when “y” is multiplied by itself → y ⤫ y, the result is 123. The number 123 has a unique non-negative square root, the principal square root.

What Is the 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

Finding the Square Root of 123

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
 

Square Root of 123 By Prime Factorization 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
 

Explore Our Programs

Square Root of 123 By Long Division Method

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 


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  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, whichis less than 23.


 Repeat this process until you reach the remainder of 0. We are left with thethe 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….
 

Square Root of 123 By Approximation

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
 

Common Mistakes and How to Avoid Them in the Square Root of 123

When we find the square root of 123, we often make some key mistakes, especially when we solve problems related to that. So, let’s see some common mistakes and their solutions.

Download Worksheets

Problem 1

Estimate the value of √123 using an initial guess of 11.08 Solution: using the formula, New Guess=(Initial Guess + (Given Number / Initial Guess))/ 2

Okay, lets begin

Applying the formula, 


New guess= (11.08 + (123/11.08))/2 


= (11.08+ 11.1010)/2


=22.181/2


 =11.0905


Again, repeating the process,


New guess= (11.0905 + (123/11.0905))/2 


= (11.0905+ 11.09057)/2


 =22.1855/2


=11.09278


hence, after a few iterations, the value of √123 is approximately 11.09278


Answer: 11.09278 approx
 

Explanation

Using the formula for New Guess, we found the approximate value of the square root 123 by repeated iterations, where New Guess=(Initial Guess + (Given Number / Initial Guess))/ 2
 

Well explained 👍

Problem 2

Find the length of a side of a square whose area is 123 cm²

Okay, lets begin

Given, the area = 123 cm2


We know that, (side of a square)2 = area of square


Or,  (side of a square)2 = 123


 Or,  (side of a square) = √123


Or, side of a square = 11.0905.


But, the length of a square is a positive quantity only, so, the length of the side is 11.0905 cm.

Answer: 11.0905 cm
 

Explanation

We know that, (side of a square)2 = area of square. Here, we are given with the area of the square, so, we can easily find out its square root because its square root is the measure of the side of the square

Well explained 👍

Problem 3

Simplify (√123 + √123) ⤫ √123

Okay, lets begin

 (√123 + √123) ⤫ √123


= (11.0905 + 11.0905) ⤫ 11.0905


= 22.181  ⤫ 11.0905


= 245.998

Answer: 245.998
 

Explanation

We first solved the part inside the brackets, i.e., √123 + √123, which resulted into 22.181, and then multiply it with √123 which is 11.0905 we get 245.998.
 

Well explained 👍

Problem 4

if x= √123, what is x²-3 ?

Okay, lets begin

x= √123


⇒ x2 = 123


⇒ x2-3 = 123-3


⇒ x2-3 = 120

Answer: 120
 

Explanation

We squared the given value of x and then subtracted 3 from it.

Well explained 👍

Problem 5

Calculate (√123/3 + √123/10)

Okay, lets begin

√123/3 + √123/10


=  11.09053/ 3 +  11.09053/10


= 3.6968 + 1.10905


= 4.80585

Answer: 4.80585
 

Explanation

From the given expression, we first found the value of the square root of 123 then solved by simple divisions and then simple addition.
 

Well explained 👍

FAQs on 123 Square Root

1.Is 123 a prime number?

No, 123 is not a prime number.

2.Is the square root of 123 a whole number?

No, 11.0905365, the square root of 123, is never a whole number

3.Is 123 a perfect square or a non-perfect square?

123 is a non-perfect square, since 123 = (11.0905365)2.

4.Is the square root of 123 a rational or irrational number?

The square root of 123 is ±11.0905365. So, 11.0905365 is an irrational number since it cannot be obtained by dividing two integers and cannot be written in the form p/q.

5.What are the factors of 123?

The principal square root of 123 is ±11.0905365, the positive value, but not -11.0905365.

6.Is the square root of 123 a real number?

Important Glossaries for Square Root of 123

  • Exponential form: 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 
  • Factorization:  Expressing the given expression as a product of its factors Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3
  • Prime Numbers:  Numbers which are greater than 1, having only 2 factors as →1 and Itself. Ex: 1,3,5,7,....
  • 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. 
  • 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.

What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math

▶

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.