Square Root of 1116

The square root is the inverse of the square of the number. 1116 is not a perfect square. The square root of 1116 is expressed in both radical and exponential form. In the radical form, it is expressed as √1116, whereas (1116)^(1/2) in the exponential form. √1116 ≈ 33.426, 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 1116 is broken down into its prime factors.

Step 1: Finding the prime factors of 1116 Breaking it down, we get 2 × 2 × 3 × 3 × 31 = 2^2 × 3^2 × 31

Step 2: Now we found out the prime factors of 1116. The second step is to make pairs of those prime factors. Since 1116 is not a perfect square, therefore the digits of the number can’t be grouped in pairs completely. Therefore, calculating √1116 using prime factorization results in an approximate value.

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 digits of 1116 from right to left. We consider 11 and 16 as groups.

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

Step 3: Now let us bring down 16, making the new dividend 216. 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 the sum of the dividend and quotient. Now we get 6n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 216. Let us consider n as 3, now 63 × 3 = 189.

Step 6: Subtract 189 from 216; the difference is 27, and the quotient is 33.

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 2700.

Step 8: Now we need to find the new divisor, which is 66.5 because 66.5 × 40 = 2660.

Step 9: Subtracting 2660 from 2700, we get the result 40.

Step 10: Now the quotient is 33.4.

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 √1116 ≈ 33.426.

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 1116 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √1116. The closest perfect square below 1116 is 1089, and the closest perfect square above 1116 is 1156. √1116 falls somewhere between 33 and 34.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using the formula (1116 - 1089) ÷ (1156 - 1089) = 27 ÷ 67 ≈ 0.403. 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 33 + 0.403 = 33.403, so the square root of 1116 is approximately 33.403.

• 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, with q not equal to zero and p and q being integers.

• Principal square root: A number has both positive and negative square roots; however, it is usually the positive square root that is used in real-world applications.

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

• Approximation method: A method to find an approximate value of the square root of a non-perfect square number by finding nearby perfect squares and estimating the decimal value.