Square Root of 124

The square root is the inverse of squaring a number. 124 is not a perfect square. The square root of 124 is expressed in both radical and exponential form. In the radical form, it is expressed as √124, whereas (124)^(1/2) in the exponential form. √124 ≈ 11.13553, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

 

The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, methods like long division and approximation are used. Let us now learn the following methods:

• Prime factorization method
  
   
• Long division method
  
   
• Approximation method

The product of prime factors is the prime factorization of a number. Now, let us look at how 124 is broken down into its prime factors:

Step 1: Finding the prime factors of 124 Breaking it down, we get 2 x 2 x 31: 2² x 31

Step 2: Now we found out the prime factors of 124. Since 124 is not a perfect square, the prime factors cannot be grouped into pairs.

Therefore, calculating the square root of 124 using prime factorization directly is not possible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step:

Step 1: To begin with, we need to group the numbers from right to left. In the case of 124, we need to group it as 24 and 1.x

Step 2: Now we need to find n whose square is 1. We can say n is ‘1’ because 1 x 1 is lesser than or equal to 1. Now the quotient is 1; after subtracting 1 - 1, the remainder is 0.

Step 3: Now let us bring down 24, which is the new dividend. Add the old divisor with the same number, 1 + 1, to get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 2n × n ≤ 24. Let us consider n as 1, then 21 x 1 = 21.

Step 6: Subtract 24 from 21; the difference is 3, and the quotient is 11.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 300.

Step 8: Now we need to find the new divisor that is 23, because 231 x 3 = 693.

Step 9: Subtracting 693 from 700, we get the result 7.

Step 10: Now the quotient is 11.3.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √124 ≈ 11.13

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 124 using the approximation method.

Step 1: Now we have to find the closest perfect square of √124. The smallest perfect square less than 124 is 121, and the largest perfect square greater than 124 is 144. √124 falls somewhere between 11 and 12.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (124 - 121) / (144 - 121) ≈ 0.13 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 11 + 0.13 = 11.13.

So the square root of 124 is approximately 11.13.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse of squaring is finding the square root, √16 = 4.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is more commonly used due to its utility in real-world situations. This is often referred to as the principal square root.
   
• Long division method: A method used to find the square root of a number, especially when the number is not a perfect square, through successive approximation.
   
• Approximation: A technique to estimate the value of a square root by identifying two perfect squares between which the number falls and using interpolation.