Square Root of 122

The square root is the inverse of the square of the number. 122 is not a perfect square. The square root of 122 is expressed in both radical and exponential form. In the radical form, it is expressed as √122, whereas (122)^(1/2) is in the exponential form. √122 ≈ 11.0454, 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, the prime factorization method is not used for non-perfect square numbers, where long-division and approximation methods 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 122 is broken down into its prime factors.

Step 1: Finding the prime factors of 122 Breaking it down, we get 2 x 61: 2¹ x 61¹

Step 2: Now we found out the prime factors of 122. The second step is to make pairs of those prime factors. Since 122 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 122 using prime factorization is impossible.

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 digits of 122 from right to left.

Step 2: Now we need to find n whose square is less than or equal to 1. We can say n as ‘1’ because 1 x 1 is less 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 22, 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 ≤ 22. Let us consider n as 1; now 2 x 1 x 1 = 2

Step 6: Subtract 22 from 2; the difference is 20, 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 zeroes to the dividend. Now the new dividend is 2000.

Step 8: Now we need to find the new divisor that is 21 because 211 x 9 = 1899

Step 9: Subtracting 1899 from 2000, we get the result 101.

Step 10: Now the quotient is 11.0

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

So, the square root of √122 ≈ 11.04

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 122 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √122. The smallest perfect square less than 122 is 121, and the largest perfect square greater than 122 is 144. √122 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, (122 - 121) ÷ (144 - 121) = 1/23 ≈ 0.0435 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 + 0.0435 ≈ 11.04.

So the square root of 122 is approximately 11.04.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
   
• Integers: The combination of whole numbers and negative numbers are integers. For example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, etc., are integers.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.