Square Root of 2108

The square root is the inverse of the square of the number. 2108 is not a perfect square. The square root of 2108 is expressed in both radical and exponential form. In radical form, it is expressed as √2108, whereas in exponential form it is expressed as (2108)^(1/2). √2108 ≈ 45.913, 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 and approximation methods 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 2108 is broken down into its prime factors.

Step 1: Finding the prime factors of 2108

Breaking it down, we get 2 x 2 x 3 x 3 x 59: 2^2 x 3^2 x 59

Step 2: Now we found the prime factors of 2108. The second step is to make pairs of those prime factors. Since 2108 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 2108 using prime factorization cannot yield an exact integer 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 2108, we need to group it as 08 and 21.

Step 2: Now we need to find n whose square is less than or equal to 21. We can say n as ‘4’ because 4 x 4 = 16 is lesser than or equal to 21. Now the quotient is 4, after subtracting 21 - 16, the remainder is 5.

Step 3: Now let us bring down 08, making the new dividend 508. Add the old divisor with the same number 4 + 4, we get 8, which will be our new divisor.

Step 4: The new divisor will be 80 (8, with a digit to find in place of zero) as we need to find a number n such that 80n × n ≤ 508.

Step 5: We find n as 6 because 806 x 6 = 483.

Step 6: Subtract 508 from 483, the difference is 25, and the quotient is 46.

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, which is 919 because 919 x 2 = 1838.

Step 9: Subtracting 1838 from 2500, we get the result 662.

Step 10: Now the quotient is 45.9.

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 √2108 is approximately 45.91.

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 2108 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2108. The smallest perfect square of 2108 is 2025, and the largest perfect square of 2108 is 2116. √2108 falls somewhere between 45 and 46.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (2108 - 2025) / (2116 - 2025) = 0.913.

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 + 0.913 = 45.913, so the square root of 2108 is approximately 45.913.

• 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.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
   
• Long division method: A method used to find the square root of numbers that are not perfect squares by dividing and finding each digit of the square root step by step.