Square Root of 123
The square root of 123 can be written as √123. This is raising 123 to the power ½. In this mini lesson, we will learn about the square root of 123 and find out whether the square root of 123 is rational or irrational. We will also learn how to find the square root of 123 by long division method.
 Square Root of 123: √123 = 11.090
 Square of 123: 123^{2} = 15,129
What Is the Square Root of 123?
The square root of a number is the inverse process of squaring a number.
 A square of a number x is y^{2 }⇒ x = y × y. For example, 123 = 11.090 × 11.090
 √123 = √(11.090 × 11.090)
 123^{½} = (11.090^{2})^{½ }
 11.090 = 123^{½} ⇒ √123 = ±11.090
Is Square Root of 123 Rational or Irrational?
√123 = 11.09053650640942
√123 cannot be written in the form of p/q, hence, it is an irrational number. The value of the square root of 123 is a decimal number whose digits after the point are neverending or nonterminating.
How to Find the Square Root of 123?
The square root of 123 or any number can be calculated in many ways. Two of the common methods are the approximation method and the long division method.
Square Root of 123 by Approximation Method
 Take two perfect square numbers. One should be just smaller than 123 and the other a little greater than 123. √121 < √123 < √144
 11 < √123 < 12
 Using the average method, divide 123 by 11 or 12
 Let us divide by 11 ⇒ 123 ÷ 11 = 11.18
 Find the average of 11.18 and 11.
 (11.18 + 11) / 2 = 22.18 ÷ 2 = 11.09
 √123 ≈ 11.09
Square Root of 123 by the Long Division Method
The long division method helps us to find a more accurate value of the square root of any number. Let's see how to find the square root of 123 by the long division method.
 Write 123 as 1 23. 00 00 00 and from the right take the numbers in pairs. 1 is left alone. Let's divide 1 first.
 Find a number × number such that it results ≤ 1. We get 1 × 1 = 1.
 Get the remainder as 0 and bring down the next pair 23 for division.
 Double the quotient obtained. It is 2 and we hav 20 as our new divisor.
 Find a (number + 20 ) × number such that it results ≤ 23. We get (number + 20 ) × number = (1 + 20) × 1 = 21.
 Subtract this from 23. We will get the remainder as 2. Bring down the next pair of zeros. 200 is considered as the dividend for the next division.
 Double the quotient obtained. It is 22 and we hav 220 as our new divisor.
 Find a (number + 220 ) × number such that it results ≤ 200. We cannot find such a number. We get (0+ 220 ) × 0.
 We will get the remainder as 200. Bring down the next pair of zeros. 20000 is the new dividend.
 The quotient we have obtained so far is 11.0 and on doubling we get 220. We take 2200 as our new divisor.
 Find a (number + 2200 ) × number such that it results ≤ 2 00 00. We get (number + 2200 ) × number = (9 + 2200 ) × 9 = 19881.
 We can repeat the process until we approximate the square root of 123 to 3 decimal places.
Explore square roots using illustrations and interactive examples:
Tips and Tricks
 The square root of 123 lies between the square root of 121 and 144. Therefore, 11 < √123 < 12.
 You can then use the average method to evaluate the approximate value of √123.
Important Notes
 The square root of 123 is 11.090 approximated to 3 decimal places.
 The simplified form of √123 in its radical form is √(41×3)
 √123 is an irrational number.
Square Root of 123 Solved Examples

Example 1: Henry wants to stack 123 books in such a way that there are as many number of books as the number of rows. How many books will he have to take out and how many books will he have to add to make such an arrangement?
Solution: He needs to find the least number to be added to 123 and subtracted from 123 to make them perfect squares. 123 lies between 121 and 144 which are the perfect squares closer to 123.
123  2 = 121 ⇒11 × 11 = 121
123 + 21 = 144 ⇒12 × 12 = 144
If he decides to have 11 books arranged in 11 rows, he will have to remove two books from the 123 books and if he decides to have 12 books arranged in 12 rows, he will need to add 21 more books to fit into the arrangement.

Example 2: Evaluate: √123 × √205
Solution: √123= √(3 × 41)
√205 = √(5 × 41)
√123 × √205 = √(3 × 41) × √(5 × 41)
= √3 × √41× √5 × √41
= √3 × √5 × √41× √41
√123 × √205 = 41√15
FAQs on Square Root of 123
What is the square root of 123?
The square root of 123 is √123 = 11.090
What is the square root of 123 simplified?
√(41×3) is the simplest form of √123
How to find the square root of 123?
The square root of 123 can be found using the long division method.
Is √123 a rational number?
√123 is an irrational number because the value of √123 is a nonterminating decimal.
How to find the square root of 123 to the nearest hundredth?
The square root of 123 is evaluated using the long division method and rounded off to the nearest hundredth. √123 = 11.09