Square Root of 2012

The square root is the inverse of the square of the number. 2012 is not a perfect square. The square root of 2012 is expressed in both radical and exponential form. In the radical form, it is expressed as √2012, whereas (2012)^(1/2) in the exponential form. √2012 ≈ 44.833, 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 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 2012 is broken down into its prime factors.

Step 1: Finding the prime factors of 2012

Breaking it down, we get 2 x 2 x 503: 2² x 503¹

Step 2: Now we found out the prime factors of 2012. The second step is to make pairs of those prime factors. Since 2012 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 2012 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 2012, we need to group it as 12 and 20.

Step 2: Now we need to find n whose square is less than or equal to 20. We can say n is '4' because 4² = 16, which is less than 20. The quotient is 4, and after subtracting 16 from 20, the remainder is 4.

Step 3: Now let us bring down 12, which is the new dividend. 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 the sum of the dividend and quotient. Now we get 8n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 8n × n ≤ 412. Let us consider n as 5, now 85 x 5 = 425, which is too high, so we consider n as 4. We have 84 x 4 = 336.

Step 6: Subtract 336 from 412, the difference is 76, and the quotient becomes 44.

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 7600.

Step 8: Now we need to find n for the new divisor 888n ≤ 7600. Let's consider n as 8, then 888 x 8 = 7104.

Step 9: Subtracting 7104 from 7600, we get the result 496.

Step 10: Now the quotient is 44.8

Step 11: Continue doing these steps until we get two numbers after the decimal point. If no decimal values appear, continue until the remainder is zero.

So the square root of √2012 is approximately 44.83.

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 2012 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √2012. The smallest perfect square less than 2012 is 2025, and the largest perfect square greater than 2012 is 1936. √2012 falls somewhere between 44 and 45.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Going by the formula (2012 - 1936) ÷ (2025 - 1936) = 76/89 ≈ 0.85

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 + 0.85 ≈ 44.85, so the square root of 2012 is approximately 44.85.

• 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, q is not equal to zero and p and q are integers.
   
• Long division method: A method used to find the square roots of non-perfect squares by dividing and averaging.
   
• Approximation method: A method used to find a close estimate of a square root by comparing with nearby perfect squares.
   
• Perfect square: A number that is the square of an integer. Example: 144, because 12 x 12 = 144.