Square Root of 1360

The square root is the inverse of the square of the number. 1360 is not a perfect square. The square root of 1360 is expressed in both radical and exponential form. In the radical form, it is expressed as √1360, whereas (1360)^(1/2) in the exponential form. √1360 ≈ 36.891, 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 1360 is broken down into its prime factors.

Step 1: Finding the prime factors of 1360

Breaking it down, we get 2 x 2 x 2 x 2 x 5 x 17: 2^4 x 5 x 17

Step 2: Now we found out the prime factors of 1360. The second step is to make pairs of those prime factors. Since 1360 is not a perfect square, therefore the digits of the number can’t be grouped in a pair. Therefore, calculating 1360 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 1360, we need to group it as 60 and 13.

Step 2: Now we need to find n whose square is 13. We can say n as '3' because 3 x 3 is lesser than or equal to 13. Now the quotient is 3; after subtracting 9 from 13, the remainder is 4.

Step 3: Now let us bring down 60, which is the new dividend. Add the old divisor with the same number: 3 + 3, we get 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 460. Let us consider n as 7, now 67 x 7 = 469, which is too large, so we use n = 6 instead, 66 x 6 = 396.

Step 6: Subtract 396 from 460; the difference is 64, and the quotient is 36.

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 6400.

Step 8: Now we need to find the new divisor that is 736 because 736 x 8 = 5888.

Step 9: Subtracting 5888 from 6400, we get the result 512.

Step 10: Now the quotient is 36.8

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

So the square root of √1360 is approximately 36.89.

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 1360 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1360. The smallest perfect square less than 1360 is 1296, and the largest perfect square greater than 1360 is 1444. √1360 falls somewhere between 36 and 38.

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 (1360 - 1296) ÷ (1444 - 1296) = 64 ÷ 148 ≈ 0.432. 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 36 + 0.432 ≈ 36.89.

So the square root of 1360 is approximately 36.89.

• 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, but in practical applications, the positive square root is often used, 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 24 is 2^3 x 3.

• Long division method: A technique used to find the square root of a non-perfect square number by dividing the number into groups and systematically finding the quotient.