Square Root of 1962

The square root is the inverse of the square of the number. 1962 is not a perfect square. The square root of 1962 is expressed in both radical and exponential form. In the radical form, it is expressed as √1962, whereas (1962)^(1/2) in the exponential form. √1962 ≈ 44.283, 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 the long-division method and approximation method 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 1962 is broken down into its prime factors:

Step 1: Finding the prime factors of 1962 Breaking it down, we get 2 x 3 x 3 x 109: 2^1 x 3^2 x 109^1

Step 2: Now we found out the prime factors of 1962. The second step is to make pairs of those prime factors. Since 1962 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 1962 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 numbers from right to left. In the case of 1962, we need to group it as 62 and 19.

Step 2: Now we need to find n whose square is 19. We can say n as ‘4’ because 4 x 4 = 16 is lesser than or equal to 19. Now the quotient is 4, and after subtracting 16 from 19, the remainder is 3.

Step 3: Now let us bring down 62, which is the new dividend. Add the old divisor with the same number, 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be 8n, and we need to find the value of n.

Step 5: The next step is finding 8n x n ≤ 362. Let us consider n as 4, now 8 x 4 = 32, and 32 x 4 = 128.

Step 6: Subtract 128 from 362, the difference is 234, and the quotient is 44.

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 23400.

Step 8: Now we need to find the new divisor. It will be 889 because 889 x 9 = 8001.

Step 9: Subtracting 8001 from 23400, we get the result 15499.

Step 10: Now the quotient is 44.2

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 √1962 is approximately 44.28.

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 1962 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1962. The smallest perfect square less than 1962 is 1936, and the largest perfect square more than 1962 is 2025. √1962 falls somewhere between 44 and 45.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (1962 - 1936) / (2025 - 1936) ≈ 0.28.

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 44 + 0.28 = 44.28.

So the square root of 1962 is approximately 44.28.

• Square root: A square root is the inverse of a square. For 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 a principal square root.

• Prime factorization: The expression of a number as a product of its prime factors.

• Long division method: A mathematical method to find the square root of a non-perfect square number by dividing it into groups and using subtraction and addition steps to approximate the square root.