Square Root of 1972

The square root is the inverse of the square of a number. 1972 is not a perfect square. The square root of 1972 is expressed in both radical and exponential form. In radical form, it is expressed as √1972, whereas (1972)^(1/2) in exponential form. √1972 ≈ 44.404, 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, the prime factorization method is not used; instead, 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 1972 is broken down into its prime factors:

Step 1: Finding the prime factors of 1972 Breaking it down, we get 2 x 2 x 13 x 19: 2² x 13 x 19

Step 2: Now we found out the prime factors of 1972. The second step is to make pairs of those prime factors. Since 1972 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √1972 using prime factorization is not feasible.

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 1972, we can group it as 72 and 19.

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

Step 3: Bring down 72, making the new dividend 372. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 8n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 372. Let us consider n as 4, now 84 x 4 = 336.

Step 6: Subtract 336 from 372. The difference is 36, 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 3600.

Step 8: Now we need to find the new divisor that is 88 because 888 x 4 = 3552.

Step 9: Subtracting 3552 from 3600, we get the result 48.

Step 10: Now the quotient is 44.4.

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

So the square root of √1972 is approximately 44.40.

The approximation method is an easy method for finding square roots. Now let us learn how to find the square root of 1972 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √1972. The smallest perfect square less than 1972 is 1936 (44²), and the largest perfect square greater than 1972 is 2025 (45²). √1972 falls somewhere between 44 and 45.

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

Going by the formula (1972 - 1936) / (2025 - 1936) ≈ 0.404

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 + 0.404 ≈ 44.404, so the square root of 1972 is approximately 44.404.

• Square root: A square root is the inverse of a square. For example, 4² = 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 is more commonly used due to its applications in the real world. This is why it is also known as the principal square root.

• Prime factorization: The process of expressing a number as the product of its prime factors. For example, the prime factorization of 1972 is 2² x 13 x 19.

• Decimal: A decimal is a number that includes a whole number and a fraction expressed in a single number, such as 7.86, 8.65, and 9.42.