Square Root of 2005

The square root is the inverse of the square of the number. 2005 is not a perfect square. The square root of 2005 is expressed in both radical and exponential form. In the radical form, it is expressed as √2005, whereas (2005)^(1/2) is in the exponential form. √2005 ≈ 44.778, 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 2005 is broken down into its prime factors.

Step 1: Finding the prime factors of 2005

Breaking it down, we get 5 x 401.

Step 2: Now we found out the prime factors of 2005. The second step is to make pairs of those prime factors. Since 2005 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 2005 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 2005, we need to group it as 05 and 20.

Step 2: Now we need to find n whose square is closest to 20. We can say n is 4 because 4 x 4 = 16, which is lesser than or equal to 20. Now the quotient is 4, and the remainder is 20 - 16 = 4.

Step 3: Now let us bring down 05, giving us a new dividend of 405. Add the old divisor with the same number: 4 + 4 = 8, which will be our new divisor.

Step 4: We need to find the largest possible digit x such that 8x * x ≤ 405. For x = 5, 85 * 5 = 425, which is more than 405. For x = 4, 84 * 4 = 336, which is less than 405.

Step 5: Subtract 336 from 405; the remainder is 69. Our quotient is now 44.

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. Now the new dividend is 6900.

Step 7: Find the new divisor, 88, by adding 4 to the quotient, making it 88x. Determine x such that 88x * x ≤ 6900. For x = 7, 887 * 7 = 6209.

Step 8: Subtracting 6209 from 6900 gives us a remainder of 691.

Step 9: The quotient is now 44.7.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √2005 is approximately 44.78.

The approximation method is another method for finding square roots, and 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 2005 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2005. The smallest perfect square less than 2005 is 1936 (44^2), and the largest perfect square more than 2005 is 2025 (45^2). √2005 falls somewhere between 44 and 45.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). (2005 - 1936) ÷ (2025 - 1936) = 69 ÷ 89 ≈ 0.775

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 44 + 0.775 ≈ 44.775, so the square root of 2005 is approximately 44.775.

• Square root: A square root is the inverse of a square. Example: 4² = 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 the 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 technique used to find the square root of a non-perfect square by repeated division and subtraction.