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: 123² = 15,129

• A square of a number x is y² ⇒ x = y × y. For example, 123 = 11.090 × 11.090
• √123 = √(11.090 × 11.090)
• 123^½ = (11.090²)^½ 
• 11.090 = 123^½ ⇒ √123 = ±11.090

√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 never-ending or non-terminating. 

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: 

• Square root of 98
• Square root of 54
• Square root of 74
• Square root of 180
• Square root of 120

• 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.

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 non-terminating 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