Square Root of 509

The square root is the inverse of the square of a number. 509 is not a perfect square. The square root of 509 is expressed in both radical and exponential form. In the radical form, it is expressed as √509, whereas (509)^(1/2) in the exponential form. √509 ≈ 22.549, 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 and approximation methods 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 509 is broken down into its prime factors.

Step 1: Finding the prime factors of 509

Breaking it down, we get 7 x 73: 7^1 x 73^1

Step 2: Now that we have found the prime factors of 509, the next step is to make pairs of those prime factors. Since 509 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √509 using prime factorization is not feasible.

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 509, we need to group it as 09 and 5.

Step 2: Now we need to find n whose square is 5. We can say n is ‘2’ because 2 x 2 = 4, which is lesser than or equal to 5. Now the quotient is 2 and after subtracting 4 from 5, 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, 2 + 2, to get 4, which will be our new divisor.

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

Step 5: The next step is finding 4n x n ≤ 109. Let us consider n as 2, now 4 x 2 x 2 = 88.

Step 6: Subtract 88 from 109, and the difference is 21.

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 2100.

Step 8: Now we need to find the new divisor, which is 45, because 445 x 5 = 2225.

Step 9: Subtracting 2225 from 2100, we get the result -125.

Step 10: Now the quotient is 22.5.

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

So the square root of √509 ≈ 22.54

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 509 using the approximation method.

Step 1: Now we have to find the closest perfect square of √509. The smallest perfect square less than 509 is 484, and the largest perfect square greater than 509 is 529. √509 falls somewhere between 22 and 23.

Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (509 - 484) / (529 - 484) = 0.56

Using the formula, we identified the decimal part of our square root. The next step is adding the value we got initially to the decimal number, which is 22 + 0.56 = 22.56, so the square root of 509 is approximately 22.56.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, 5^2 = 25, and the inverse of the square is the square root, that is √25 = 5.
   
• 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.
   
• Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, 2, 3, 5, 7, etc.
   
• Long division method: A traditional method used to find the square root of a non-perfect square number by systematically dividing and averaging.
   
• Approximation method: A method used to estimate the square root of a number by identifying the closest perfect squares and calculating a fractional approximation.