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120 Learners

Last updated on ** September 18th, 2024**

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.

The square root of 123 is ±11.0905365, where 11.0905365 is the positive solution of the equation x^{2} = 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}

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

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

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 1

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

quotient, and we get a remainder. Double the divisor 1, we get 2, and then the

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, which

is less than 23.

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

the 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….

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

Exponential form

1) 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

2) Factorization

Expressing the given expression as a product of its factors Ex: 48=2 ⤬ 2 ⤬ 2 ⤬ 2 ⤬ 3

3) Prime Numbers

Numbers which are greater than 1, having only 2 factors as →1 and Itself.

Ex: 1,3,5,7,....

4) 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.

5) 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.