Square Root of 1311

The square root is the inverse of the square of the number. 1311 is not a perfect square. The square root of 1311 is expressed in both radical and exponential form. In radical form, it is expressed as √1311, whereas (1311)^(1/2) in exponential form. √1311 ≈ 36.213, 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 1311 is broken down into its prime factors:

Step 1: Finding the prime factors of 1311

Breaking it down, we get 3 x 3 x 3 x 3 x 2 x 17: 3^3 x 2^1 x 17^1

Step 2: Now we found out the prime factors of 1311. The second step is to make pairs of those prime factors. Since 1311 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 1311 using prime factorization is not straightforward.

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 1311, we need to group it as 11 and 13.

Step 2: Now, we need to find n whose square is less than or equal to 13. We can say n as ‘3’ because 3^2 = 9 is less 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 11, making the new dividend 411. 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 60 (6 plus a digit to be guessed next). Now we need to find a number such that (60 + n) × n ≤ 411.

Step 5: Let us consider n as 6, so 66 × 6 = 396.

Step 6: Subtract 396 from 411; the difference is 15, 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 1500.

Step 8: Now, we need to find the new divisor, which will be 720 because 722 × 2 = 1444.

Step 9: Subtracting 1444 from 1500, we get the result 56.

Step 10: Now the quotient is 36.2.

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 √1311 is approximately 36.21.

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 1311 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1311. The smallest perfect square below 1311 is 1296 (36^2), and the next perfect square is 1369 (37^2). √1311 falls somewhere between 36 and 37.

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 (1311 - 1296) ÷ (1369 - 1296) = 15 ÷ 73 ≈ 0.205 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 + 0.205 = 36.205, so the square root of 1311 is approximately 36.21.

• Square root: A square root is the inverse operation of squaring a number. For example, 6^2 = 36, and the inverse of the square is the square root, that is √36 = 6.

• 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.

• Prime factorization: The process of expressing a number as the product of its prime factors.

• Decimal: A number that contains a whole number and a fractional part separated by a decimal point. For example, 3.142, 36.21, and 0.5 are decimals.

• Long division method: A method used to divide large numbers into simpler steps to find the quotient and remainder, often used to calculate square roots for non-perfect squares.