Square Root of 11.6

The square root is the inverse of the square of the number. 11.6 is not a perfect square. The square root of 11.6 is expressed in both radical and exponential form. In the radical form, it is expressed as √11.6, whereas (11.6)^(1/2) in the exponential form. √11.6 ≈ 3.405877, 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 long-division method and approximation method are used. Let us now learn the following methods:

• Long division method
• Approximation method

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 11.6, we need to consider it as 11.60.

Step 2: Now we need to find n whose square is less than or equal to 11. We can say n as ‘3’ because 3 x 3 = 9 is less than 11. Now the quotient is 3, and after subtracting 9 from 11, the remainder is 2.

Step 3: Now let us bring down 60, making the new dividend 260. Add the old divisor with the same number 3 + 3 to get 6, which will be our new divisor.

Step 4: The new divisor will be 6n. We need to find the value of n such that 6n x n ≤ 260.

Step 5: Let us consider n as 4, now 64 x 4 = 256.

Step 6: Subtract 256 from 260, the difference is 4, and the quotient is 3.4.

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 400.

Step 8: Now we need to find the new divisor. It will be 68 because 684 x 4 = 2736, and 68 x 5 = 340, which is less than 400.

Step 9: Subtracting 340 from 400 gives the result 60.

Step 10: Now the quotient is approximately 3.40.

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

So the square root of √11.6 ≈ 3.405877.

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 11.6 using the approximation method.

Step 1: Now we have to find the closest perfect square of √11.6. The smallest perfect square less than 11.6 is 9, and the largest perfect square greater than 11.6 is 16. √11.6 falls somewhere between 3 and 4.

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 (11.6 - 9) / (16 - 9) = 2.6 / 7 ≈ 0.371428.

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 3 + 0.405877 ≈ 3.405877, so the square root of 11.6 is approximately 3.405877.

• 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.
   
• Approximation: The process of finding a value that is close enough to the correct answer, usually with some sort of calculation or estimation.
   
• Long division method: A method used to find the square root of non-perfect square numbers by dividing the number into pairs and finding the root step by step.
   
• Rational number: A rational number is a number that can be expressed as a fraction p/q, where q is not zero, and p and q are integers.