11254 LearnersLast updated on May 26th, 2025
Square Root of 123
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
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.
Mistake 1
Misunderstanding symbol
Often when √123 is mistaken as 1232, we square the number 123 and get the result as 15129. So, understanding of symbols should be clear.
Square Root of 123 Examples
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
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
Problem 2
Find the length of a side of a square whose area is 123 cm²
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
Problem 3
Simplify (√123 + √123) ⤫ √123
(√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.
Problem 4
if x= √123, what is x²-3 ?
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.
Problem 5
Calculate (√123/3 + √123/10)
√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.
FAQs on 123 Square Root
1.Is 123 a prime number?
2.Is the square root of 123 a whole number?
3.Is 123 a perfect square or a non-perfect square?
4.Is the square root of 123 a rational or irrational number?
5.What are the factors of 123?
6.Is the square root of 123 a real number?
7.How does learning Algebra help students in United States make better decisions in daily life?
8.How can cultural or local activities in United States support learning Algebra topics such as Square Root of 123?
9.How do technology and digital tools in United States support learning Algebra and Square Root of 123?
10.Does learning Algebra support future career opportunities for students in United States?
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.
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