Square Root of 1183

The square root is the inverse of the square of the number. 1183 is not a perfect square. The square root of 1183 is expressed in both radical and exponential form. In the radical form, it is expressed as √1183, whereas (1183)^(1/2) in the exponential form. √1183 ≈ 34.39882, 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 1183 is broken down into its prime factors.

Step 1: Finding the prime factors of 1183. Breaking it down, we get 1183 = 7 x 169, which further breaks down to 7 x 13 x 13: 7^1 x 13^2.

Step 2: Now we have found the prime factors of 1183. The second step is to make pairs of those prime factors. Since 1183 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 1183 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 1183, we need to group it as 83 and 11.

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, 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: Now let us bring down 83, which is the new dividend. 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, where we need to find the value of n such that 6n x n is less than or equal to 283. Let n be 4, so 64 x 4 = 256.

Step 5: Subtract 256 from 283, the difference is 27, and the quotient is 34.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 2700.

Step 7: Now we need to find the new divisor, 688, such that 688 x 4 = 2752.

Step 8: Subtracting 2752 from 2700 gives us a negative number, so we adjust n to 3. Thus, 683 x 3 = 2049.

Step 9: Subtracting 2049 from 2700, we get the result as 651.

Step 10: Now the quotient is 34.3.

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

So the square root of √1183 is approximately 34.40.

The 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 1183 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1183. The smallest perfect square less than 1183 is 1156, and the largest perfect square more than 1183 is 1225. √1183 falls somewhere between 34 and 35.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1183 - 1156) ÷ (1225 - 1156) = 27 ÷ 69 ≈ 0.3913. 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 34 + 0.3913 = 34.3913, so the square root of 1183 is approximately 34.40.

• 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, √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 the real world. That is why it is also known as the principal square root.

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

• Long division method: A step-by-step method for finding the square root of a non-perfect square number.