Square Root of 1136

The square root is the inverse of the square of the number. 1136 is not a perfect square. The square root of 1136 is expressed in both radical and exponential form. In the radical form, it is expressed as √1136, whereas (1136)^(1/2) in the exponential form. √1136 ≈ 33.694, 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 and approximation methods 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 1136 is broken down into its prime factors.

Step 1: Finding the prime factors of 1136 Breaking it down, we get 2 x 2 x 2 x 2 x 71: 2^4 x 71

Step 2: Now we found out the prime factors of 1136. The second step is to make pairs of those prime factors. Since 1136 is not a perfect square, the digits of the number can’t be grouped into pairs to result in a whole number square root. Therefore, calculating 1136 using prime factorization doesn't yield an integer.

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, group the numbers from right to left. In the case of 1136, we need to group it as 11 and 36.

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

Step 3: Bring down the next pair, 36, making the new dividend 236. Add the old divisor (3) with itself (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. We get 6n as the new divisor; we need to find the value of n.

Step 5: Find 6n × n ≤ 236. Let us consider n as 3, now 63 × 3 = 189.

Step 6: Subtract 189 from 236, the difference is 47, and the quotient is 33.

Step 7: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. The new dividend is 4700.

Step 8: Now find the new divisor. It will be 67 because 673 × 3 = 2019.

Step 9: Subtracting 2019 from 4700, we get the result 2681.

Step 10: Continue doing these steps until we get two numbers after the decimal point or until the remainder is zero. So the square root of √1136 ≈ 33.694.

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 1136 using the approximation method.

Step 1: Find the closest perfect squares around √1136. The closest perfect square less than 1136 is 1089 (33^2), and the closest perfect square greater than 1136 is 1156 (34^2). √1136 falls between 33 and 34.

Step 2: Use the formula: (Given number - smaller perfect square) / (Larger perfect square - smaller perfect square). Using the formula (1136 - 1089) ÷ (1156 - 1089) = 47 ÷ 67 ≈ 0.701. Adding 33 + 0.701 = 33.701, so the square root of 1136 is approximately 33.701.

• 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 has more prominence due to its uses in real-world applications. Hence, it is known as the principal square root.

• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 18 is 2 x 3^2.

• Long division method: A method used to find the square root of non-perfect squares by dividing the number into pairs of digits from right to left and performing division similar to long division of numbers.