Square Root of 1009

The square root is the inverse of the square of the number. 1009 is not a perfect square. The square root of 1009 is expressed in both radical and exponential form. In the radical form, it is expressed as √1009, whereas (1009)^(1/2) in the exponential form. √1009 ≈ 31.76476, 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. However, 1009 is a prime number and cannot be broken down into smaller prime factors. Therefore, calculating 1009 using prime factorization is not applicable.

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 1009, we need to group it as 09 and 10.

Step 2: Now we need to find n whose square is 10. We can approximate n as ‘3’ because 3^2 = 9 is lesser than or equal to 10. Now the quotient is 3 after subtracting 10 - 9 the remainder is 1.

Step 3: Now let us bring down 09 which is the new dividend. Add the old divisor with the same number 3 + 3 we get 6 which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 6n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 6n × n ≤ 109 let us consider n as 1, now 61 x 1 = 61.

Step 6: Subtract 109 from 61; the difference is 48, and the quotient is 31.

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 4800.

Step 8: Now we need to find the new divisor that is 637 because 637 × 7 = 4459.

Step 9: Subtracting 4459 from 4800 we get the result 341.

Step 10: Now the quotient is 31.7.

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 √1009 is approximately 31.76.

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 1009 using the approximation method.

Step 1: Now we have to find the closest perfect square of √1009. The smallest perfect square less than 1009 is 961 (31^2) and the largest perfect square greater than 1009 is 1024 (32^2). √1009 falls somewhere between 31 and 32.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (1009 - 961) ÷ (1024 - 961) = 48 ÷ 63 ≈ 0.76. 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 31 + 0.76 = 31.76.

So the square root of 1009 is approximately 31.76.

• 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.
   
• Prime number: A prime number is a number greater than 1 that has no divisors other than 1 and itself.
   
• Decimal approximation: A decimal approximation is a method of finding a close decimal value of a non-integer result, often used for irrational numbers.