Square Root of 1065

The square root is the inverse of the square of the number. 1065 is not a perfect square. The square root of 1065 is expressed in both radical and exponential form. In the radical form, it is expressed as √1065, whereas (1065)^(1/2) in the exponential form. √1065 ≈ 32.619, 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, the prime factorization method is not used; instead, 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 1065 is broken down into its prime factors.

Step 1: Finding the prime factors of 1065 Breaking it down, we get 3 x 5 x 71: 3^1 x 5^1 x 71^1

Step 2: Now we found the prime factors of 1065. The second step is to make pairs of those prime factors. Since 1065 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating 1065 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 1065, we need to group it as 65 and 10.

Step 2: Now we need to find n whose square is less than or equal to 10. We can say n is ‘3’ because 3 x 3 = 9, which is lesser than or equal to 10. Now the quotient is 3, and after subtracting 9 from 10, the remainder is 1.

Step 3: Now let us bring down 65, which is the new dividend. Add the old divisor with the same number 3 + 3, we 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; we need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 165. Let us consider n as 2; now 62 x 2 = 124.

Step 6: Subtract 165 from 124; the difference is 41, and the quotient is 32.

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 4100.

Step 8: Now we need to find the new divisor that is 649 because 649 x 6 = 3894.

Step 9: Subtracting 3894 from 4100, we get the result 206.

Step 10: Now the quotient is 32.6.

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

So the square root of √1065 is approximately 32.62.

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 1065 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1065. The smallest perfect square less than 1065 is 1024, and the largest perfect square greater than 1065 is 1089. √1065 falls somewhere between 32 and 33.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (1065 - 1024) / (1089 - 1024) = 41 / 65 ≈ 0.63. 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 + 0.63 = 32.63.

So the square root of 1065 is approximately 32.63.

• 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, which 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 the principal square root.
   
• Long division method: A method of dividing large numbers to find a quotient, often used to find the square root of non-perfect squares.
   
• Approximation method: A method used to find an approximate value of a square root by identifying perfect squares around the given number.