Square Root of 1960

The square root is the inverse of the square of the number. 1960 is not a perfect square. The square root of 1960 is expressed in both radical and exponential form. In the radical form, it is expressed as √1960, whereas (1960)^(1/2) in the exponential form. √1960 ≈ 44.2719, 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 1960 is broken down into its prime factors.

Step 1: Finding the prime factors of 1960. Breaking it down, we get 2 x 2 x 2 x 5 x 7 x 7: 2^3 x 5^1 x 7^2.

Step 2: Now we found out the prime factors of 1960. The second step is to make pairs of those prime factors. Since 1960 is not a perfect square, the digits of the number can’t be fully grouped in pairs.

Therefore, calculating 1960 using prime factorization yields an approximation.

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 1960, we need to group it as 60 and 19.

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

Step 3: Now let us bring down 60, which is the new dividend. Add the old divisor with the same number 4 + 4, we get 8, which will be our new divisor.

Step 4: The new divisor will be the sum of the current quotient times 2, which is 8.5, and we need to find the value of n.

Step 5: Finding 8n x n ≤ 360, let us consider n as 4, now 84 x 4 = 336.

Step 6: Subtract 336 from 360; the difference is 24, and the quotient becomes 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 2400.

Step 8: Now we need to find the new divisor, which is 884, because 884 x 2 = 1768.

Step 9: Subtracting 1768 from 2400, we get the remainder 632.

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 till the remainder is zero.

So the square root of √1960 is approximately 44.27.

The approximation method is another method for finding the 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 1960 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1960. The smallest perfect square less than 1960 is 1936, and the largest perfect square greater than 1960 is 2025. √1960 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).

Going by the formula (1960 - 1936) / (2025 - 1936) = 24 / 89 ≈ 0.27. Using the formula, we identified the decimal point of our square root.

The next step is adding the value we found initially to the decimal number, which is 44 + 0.27 = 44.27, so the square root of 1960 is approximately 44.27.

• 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 is more commonly used due to its applicability in real-world scenarios. This is why it is also known as the principal square root.

• Prime factorization: Prime factorization involves expressing a number as a product of its prime numbers. For example, the prime factorization of 84 is 2 x 2 x 3 x 7.

• Long division method: This is a method used to find the square root of non-perfect squares by dividing the number into pairs of digits from the right and performing division, multiplication, and subtraction repeatedly.