Square Root of 1035

The square root is the inverse of the square of a number. 1035 is not a perfect square. The square root of 1035 is expressed in both radical and exponential form. In the radical form, it is expressed as √1035, whereas (1035)^(1/2) in the exponential form. √1035 ≈ 32.187, 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, for non-perfect square numbers, methods like 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 1035 is broken down into its prime factors.

Step 1: Finding the prime factors of 1035 Breaking it down, we get 3 × 5 × 7 × 11 = 3^1 × 5^1 × 7^1 × 11^1

Step 2: Now we found out the prime factors of 1035. The second step is to make pairs of those prime factors. Since 1035 is not a perfect square, therefore the digits of the number can’t be grouped into pairs.

Therefore, calculating √1035 using prime factorization alone is not feasible.

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square numbers 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 1035, we need to group it as 35 and 10.

Step 2: Now we need to find n whose square is ≤ 10. We can choose n as ‘3’ because 3 × 3 = 9, which is less than 10. The quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Bring down 35, making the new dividend 135. Add the old divisor (3) with the same number (3), which gives us 6 as the new divisor.

Step 4: The new divisor will be 6n, where n is the new digit in the quotient. We need to find n such that 6n × n ≤ 135. Let us consider n as 2, so 62 × 2 = 124.

Step 5: Subtract 124 from 135, resulting in a remainder of 11. The quotient is now 32.

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 zeroes to the dividend. The new dividend is 1100.

Step 7: Find the new divisor, which is 644, as 644 × 1 = 644.

Step 8: Subtracting 644 from 1100 gives a remainder of 456.

Step 9: Continue the process until we achieve the desired decimal places.

The square root of 1035 ≈ 32.187.

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 1035 using the approximation method.

Step 1: Find the closest perfect squares to 1035. The smallest perfect square less than 1035 is 1024, and the largest perfect square greater than 1035 is 1089. √1035 lies between 32 and 33.

Step 2: Apply the formula (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Using the formula (1035 - 1024) ÷ (1089 - 1024) = 11 ÷ 65 ≈ 0.1692. Adding the result to the smaller square root, we get 32 + 0.1692 ≈ 32.1692.

So the square root of 1035 is approximately 32.1692.

• 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, the positive square root is often used in real-world applications, known as the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime factors.
   
• Long division method: A technique used to find the square root of non-perfect squares by dividing and averaging over several iterations.