Square Root of 1105

The square root is the inverse of the square of the number. 1105 is not a perfect square. The square root of 1105 is expressed in both radical and exponential form. In the radical form, it is expressed as √1105, whereas (1105)^(1/2) in the exponential form. √1105 ≈ 33.237, 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:

• 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 1105 is broken down into its prime factors:

Step 1: Finding the prime factors of 1105 Breaking it down, we get 5 x 13 x 17.

Step 2: Now we found out the prime factors of 1105. The second step is to make pairs of those prime factors. Since 1105 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 1105 using prime factorization is not straightforward for finding the square root.

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 1105, we need to group it as 05 and 11.

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

Step 3: Now let us bring down 05, which is the new dividend. Add the old divisor with the same number, 3 + 3 = 6, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 60n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 60n x n ≤ 205. Let us consider n as 3, now 60 x 3 = 180.

Step 6: Subtract 205 from 180, the difference is 25, 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 2500.

Step 8: Now we need to find the new divisor that is 66 because 663 x 3 = 1989.

Step 9: Subtracting 1989 from 2500, we get the result 511.

Step 10: Now the quotient is 33.2

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

So the square root of √1105 is approximately 33.23.

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 1105 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1105. The smallest perfect square less than 1105 is 1024 (32²) and the largest perfect square greater than 1105 is 1156 (34²). √1105 falls somewhere between 32 and 34.

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 (1105 - 1024) ÷ (1156 - 1024) = 81 ÷ 132 ≈ 0.614. 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 32 + 0.614 ≈ 32.614, so the square root of 1105 is approximately 33.237.

• Square root: A square root is the inverse of a square. Example: 4² = 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 the real world. That is the reason it is also known as a principal square root.

• Prime factorization: The process of breaking down a number into its prime factors. For example, the prime factorization of 18 is 2 x 3 x 3.

• Long division method: A method used to find the square root of numbers that are not perfect squares, involving a series of division steps to approximate the square root value.