Square Root of 609

The square root is the inverse of the square of the number. 609 is not a perfect square. The square root of 609 is expressed in both radical and exponential form. In the radical form, it is expressed as √609, whereas (609)^(1/2) in the exponential form. √609 ≈ 24.65766, 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 609 is broken down into its prime factors.

Step 1: Finding the prime factors of 609 Breaking it down, we get 3 x 7 x 29: 3^1 x 7^1 x 29^1

Step 2: Now we found out the prime factors of 609. The second step is to make pairs of those prime factors. Since 609 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 609 using prime factorization is not feasible for finding the exact square root.

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 609, we need to group it as 09 and 6.

Step 2: Now we need to find n whose square is closest to 6. We can say n as ‘2’ because 2 x 2 is 4 which is less than 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.

Step 3: Now let us bring down 09, making the new dividend 209. Add the old divisor with the same number, 2 + 2, we get 4, which will be our new divisor.

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

Step 5: The next step is finding 4n x n ≤ 209. Let us consider n as 5, then 45 x 5 = 225, which is too large. Try n as 4, then 44 x 4 = 176.

Step 6: Subtract 176 from 209; the difference is 33, and the quotient is 24.

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 3300.

Step 8: Now we need to find the new divisor, which is 489 because 489 x 6 = 2934.

Step 9: Subtracting 2934 from 3300, we get the result 366.

Step 10: Now the quotient is 24.6.

Step 11: 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 √609 is approximately 24.66.

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 609 using the approximation method.

Step 1: Now we have to find the closest perfect square of √609. The smallest perfect square of 609 is 576 (24^2), and the largest perfect square of 609 is 625 (25^2). √609 falls somewhere between 24 and 25.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (609 - 576) ÷ (625 - 576) = 33 ÷ 49 ≈ 0.6735. 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 24 + 0.6735 ≈ 24.67, so the square root of 609 is approximately 24.67.

• 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, q is not equal to zero, and p and q are integers.

• Perfect square: A perfect square is a number that is the square of an integer. For example, 16 is a perfect square because it is 4^2.

• Long division method: A method used to find the square root of non-perfect square numbers by repeatedly dividing and averaging.

• Approximation method: A method used to estimate the square root of a number by finding its closest perfect squares and using them to approximate the value.