Square Root of 123 + Solution With Free Steps

The square root of 123 can be expressed as √123, equivalent to y, where the symbol √ is called the square root. The square root of a number produces the number that, when multiplied by itself, produces the actual number. For instance, the square root of 123 is 11.09, which means 11.09 x 11.09 = 123. 

In this article, we will analyze and find the square root of 123 using various mathematical techniques such as the approximation and long division methods.

What Is the Square Root Of 123?

The square root of the number 123 is 11.09.

Figure 1 – Basic Definition of Square root

The square root can be defined as the quantity that can be doubled to produce the square of that similar quantity. In simple words, it can be explained as:

√123 = √(11.09 x 11.09)

√123 = √(11.09)$^2$

√123 = ±11.09

The square can be canceled with the square root as it is equivalent to 1/2; therefore, obtaining y. Hence 11.09 is 123’s square root. The square root generates both positive and negative integers.

How To Calculate the Square Root of 123?

You can calculate the square root of 123 using any of two vastly used techniques in mathematics; one is the Approximation technique, and the other is the Long Division method.

The symbol √ is interpreted as 123 raised to the power 1/2. So any number, when multiplied by itself, produces its square, and when the square root of any squared number is taken, it produces the actual number.

Let us discuss each of them to understand the concepts better.

Square Root by Long Division Method

The process of long division is one of the most common methods used to find the square roots of a given number. It is easy to comprehend and provides more reliable and accurate answers. The long division method reduces a multi-digit number to its equal parts.

Learning how to find the square root of a number is easy with the long division method. All you need are five primary operations- divide, multiply, subtract, bring down or raise, then repeat.

Following are the simple steps that must be followed to find the square root of 123 using the long division method:

Step 1

First, write the given number 123 in the division symbol, as shown in figure 1.

Step 2

Starting from the right side of the number, divide the number 123 into pairs such as 23 and 1.

Step 3

Now divide the digit a by a number, giving a number either 1 or less than 1. Therefore, the remainder is zero in this case, whereas the quotient is 1.

Step 4

After this, bring down the next pair, 23. Now the dividend is 23. To find the next divisor, we need to double our quotient obtained before. Doubling 1 gives 2; hence consider it as the next divisor.

Step 5

Now pair 2 with another number to make a new divisor that results in $\leq$ 23 when multiplied with the divisor. If the number is not a perfect square, add pair of zeros to the right of the number before starting division.

Step 6

By adding 1 to the divisor and multiplying 21 with 1 results in 21 $\leq$ 23. The remainder obtained is 2. Move the next pair of zeros that were added to the dividend down, and repeat the same process mentioned above.

Step 7

Keep on repeating the same steps till the zero remainder is obtained or if the division process continues infinitely, solve to two decimal places.

Step 8

The resulting quotient of 11.09 is the square root of 123. Figure 1 given below shows the long division process in detail:

Figure 2 – Square root of 123

Square Root by Approximation Method

The approximation method involves guessing the square root of the non-perfect square number by dividing it by the perfect square lesser or greater than that number and taking the average.

The detailed steps must be followed to find the square root of 123 using the approximation technique.

Step 1

Consider a perfect square number 121 less than 123.

Step 2

Now divide 123 by √121.

123 ÷ 11 = 11.18

Step 3

Now take the average of 11 and 11.18. The resulting number is approximately equivalent to the square root of 123, which is 11.09.

(11 + 11.18) ÷ 2 = 11.09

Figure 3 – These 2 numbers are the Square and Square root of each other

Important points

• The number 123 is not a perfect square.
• The number 123 is a rational number.
• The number 123 can be split into its prime factorization.

Is Square Root of 123 a Perfect Square?

The number 123 is not a perfect square. A number is a perfect square if it splits into two equal parts or identical whole numbers. If a number is a perfect square, it is also rational.

A number expressed in p/q form is called a rational number. All the natural numbers are rational. A square root of a perfect square is a whole number; therefore, a perfect square is a rational number.

A number that is not a perfect square is irrational as it is a decimal number. As far as 123 is concerned, it is not a perfect square. It can be proved as below:

Factorization of 123 results in 3 x 41.

Figure 4 – 123 is not a Perfect Square

Taking the square root of the above expression gives:

= √(3 x 41)

= (3 x 41)$^{1/2}$

= 11.09

This shows that 123 is not a perfect square as it has decimal places; hence it is an irrational number.

Therefore the above discussion proves that the square root of 123 is equivalent to 11.09.

Images/mathematical drawings are created with GeoGebra.

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