Square Root of 2007

The square root is the inverse of the square of the number. 2007 is not a perfect square. The square root of 2007 is expressed in both radical and exponential form. In the radical form, it is expressed as √2007, whereas (2007)^(1/2) in the exponential form. √2007 ≈ 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 2007 is broken down into its prime factors.

Step 1: Finding the prime factors of 2007

Breaking it down, we get 3 x 3 x 223: 3^2 x 223

Step 2: Now we found out the prime factors of 2007. The second step is to make pairs of those prime factors. Since 2007 is not a perfect square, therefore the digits of the number can’t be grouped in pair. Therefore, calculating 2007 using prime factorization is not straightforward.

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 2007, we need to group it as 07 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 4^2 = 16 is less than 20. The quotient is 4 after subtracting 20 - 16, the remainder is 4.

Step 3: Now let us bring down 07, making the new dividend 407. Add the old divisor with the same number 4 + 4, we get 8, which will be our new divisor.

Step 4: The next step is finding 8n × n ≤ 407. Let us consider n as 5, now 85 × 5 = 425, which is more than 407. Try n as 4, 84 × 4 = 336, which fits.

Step 5: Subtract 407 from 336, the difference is 71.

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 zeros to the dividend. Now the new dividend is 7100.

Step 7: Now we need to find the new divisor. Try 889 × 9 = 8001, too high. Try 888 × 8 = 7104, too high. Try 887 × 8 = 7096, which fits.

Step 8: Subtract 7100 from 7096, we get the result 4.

Step 9: Now the quotient is 44.8

Step 10: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √2007 is approximately 44.82.

The approximation method is another method for finding the 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 2007 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √2007. The smallest perfect square below 2007 is 1936 and the largest perfect square above 2007 is 2025. √2007 falls somewhere between 44 and 45.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula: (2007 - 1936) / (2025 - 1936) = 71 / 89 ≈ 0.7978 Adding this to 44 gives approximately 44.80, so the square root of 2007 is approximately 44.80.

• 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 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.
   
• Prime factorization: This is the process of breaking down a number into its prime factors. For example, the prime factorization of 2007 is 3 × 3 × 223.