Square Root of 2028

The square root is the inverse of the square of the number. 2028 is not a perfect square. The square root of 2028 is expressed in both radical and exponential form. In the radical form, it is expressed as √2028, whereas (2028)^(1/2) in the exponential form. √2028 ≈ 45.025, 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 2028 is broken down into its prime factors:

Step 1: Finding the prime factors of 2028

Breaking it down, we get 2 × 2 × 3 × 13 × 13: 2^2 × 3^1 × 13^2

Step 2: Now we found out the prime factors of 2028. The second step is to make pairs of those prime factors. Since 2028 is not a perfect square, the digits of the number can’t be grouped in a perfect pair. Therefore, calculating 2028 using prime factorization is possible but results in an irrational number.

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 2028, we need to group it as 28 and 20.

Step 2: Now we need to find n whose square is 20. We can say n as '4' because 4 × 4 = 16, which is less than 20. Now the quotient is 4, and after subtracting 16 from 20, the remainder is 4.

Step 3: Now let us bring down 28, making it 428 as the new dividend. Add 4 to the old divisor to get 8, which will be our new divisor.

Step 4: Find a digit n such that 8n × n ≤ 428. Let us consider n as 5, now 85 × 5 = 425.

Step 5: Subtract 425 from 428, the difference is 3, and the quotient becomes 45.

Step 6: Since the remainder is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 300.

Step 7: The new divisor is 90 because 901 × 1 = 90.

Step 8: Subtracting 90 from 300, we get the result 210.

Step 9: Now the quotient is 45.0. Continue these steps until we get two numbers after the decimal point or until the remainder is zero.

So the square root of √2028 ≈ 45.025.

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 2028 using the approximation method.

Step 1: Find the closest perfect square of √2028. The smallest perfect square less than 2028 is 2025 (which is 45^2), and the next perfect square is 2116 (which is 46^2). √2028 falls somewhere between 45 and 46.

Step 2: Now we apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (2028 - 2025) ÷ (2116 - 2025) = 3 ÷ 91 ≈ 0.033.

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 45 + 0.033 = 45.033, so the square root of 2028 is approximately 45.033.

• 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, 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.
   
• Perfect square: A perfect square is a number that can be expressed as the product of an integer with itself. For example, 9 is a perfect square because it can be written as 3 × 3.
   
• Long division method: A technique used to find the square root of a number, especially when the number is not a perfect square, involving iterative steps of division and subtraction.