Square Root of 2010

The square root is the inverse of the square of the number. 2010 is not a perfect square. The square root of 2010 is expressed in both radical and exponential form. In the radical form, it is expressed as √2010, whereas (2010)^(1/2) in the exponential form. √2010 ≈ 44.822, 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 2010 is broken down into its prime factors.

Step 1: Finding the prime factors of 2010

Breaking it down, we get 2 × 3 × 5 × 67: 2^1 × 3^1 × 5^1 × 67^1

Step 2: Now that we have found the prime factors of 2010, the second step is to make pairs of those prime factors. Since 2010 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 2010 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 2010, we need to group it as 10 and 20.

Step 2: Now we need to find n whose square is less than or equal to 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 10, 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 ≤ 410. Let us consider n as 5, now 85 × 5 = 425, which is too large, so n should be 4.

Step 6: Subtract 340 from 410; the difference is 70, and the quotient is 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 zeros to the dividend. Now the new dividend is 7000.

Step 8: Now we need to find the new divisor that is 448. Because 448 × 8 = 3584, which is too large, so we go with 447 × 7 = 3129.

Step 9: Subtracting 3129 from 7000, we get the result 3871.

Step 10: Now the quotient is 44.8.

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

So the square root of √2010 is approximately 44.82.

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 2010 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2010.

The smallest perfect square less than 2010 is 2025, and the largest perfect square less than 2010 is 1936. √2010 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)

Going by the formula (2010 - 1936) ÷ (2025 - 1936) = 74 ÷ 89 ≈ 0.8315 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 + 0.83 = 44.83, so the square root of 2010 is approximately 44.83.

• 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.
   
• Principal square root: A number has both positive and negative square roots; however, it is always a 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 is the square of an integer. Example: 16 is a perfect square because it is 4².
   
• 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.