Square Root of 1066

The square root is the inverse of the square of the number. 1066 is not a perfect square. The square root of 1066 is expressed in both radical and exponential form. In the radical form, it is expressed as √1066, whereas (1066)^(1/2) in the exponential form. √1066 ≈ 32.64965, 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 1066 is broken down into its prime factors:

Step 1: Finding the prime factors of 1066. Breaking it down, we get 2 x 533: 2^1 x 533^1 (since 533 is a prime number).

Step 2: Now we found out the prime factors of 1066. The second step is to make pairs of those prime factors. Since 1066 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 1066 using prime factorization is impossible.

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 1066, we need to group it as 66 and 10.

Step 2: Now we need to find n whose square is ≤ 10. We can say n as ‘3’ because 3 x 3 = 9, which is less 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 66, making the new dividend 166. 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 6n. We need to find the value of n.

Step 5: The next step is finding 6n x n ≤ 166. Let's consider n as 2, and 62 x 2 = 124.

Step 6: Subtract 124 from 166; the difference is 42, 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 4200.

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

Step 9: Subtracting 3894 from 4200, we get the result 306.

Step 10: Now the quotient is 32.6.

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

So, the square root of √1066 ≈ 32.65.

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 1066 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1066. The smallest perfect square less than 1066 is 1024, and the largest perfect square greater than 1066 is 1089. √1066 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). Using the formula, (1066 - 1024) ÷ (1089 - 1024) = 42 ÷ 65 ≈ 0.65. Using the formula, we identified the decimal point of our square root. The next step is adding the value we initially had to the decimal number, which is 32 + 0.65 = 32.65.

So the square root of 1066 is approximately 32.65.

• Square root: A square root is the inverse of squaring a number. Example: 5² = 25, and the inverse of the square is the square root, which is √25 = 5.
   
• 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 is used in real-world applications. That is why it is also known as the principal square root.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. Example: 2, 3, 5, 7, etc.
   
• Decimal approximation: When a number cannot be expressed exactly, it is often rounded to a decimal for practical purposes. Example: √2 ≈ 1.414.