Square Root of 2409

The square root is the inverse of squaring a number. 2409 is not a perfect square. The square root of 2409 is expressed in both radical and exponential form. In radical form, it is expressed as √2409, whereas (2409)^(1/2) in exponential form. √2409 ≈ 49.085, 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, for non-perfect square numbers, the long division method and the 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 2409 is broken down into its prime factors.

Step 1: Finding the prime factors of 2409 Breaking it down, we get 3 x 3 x 7 x 7 x 17: 3² x 7² x 17

Step 2: Now we found out the prime factors of 2409. Since 2409 is not a perfect square, therefore, calculating √2409 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 2409, we need to group it as 24 and 09.

Step 2: Now we need to find n whose square is less than or equal to 24. We can say n is ‘4’ because 4 x 4 = 16, which is less than or equal to 24. Subtracting, we get 24 - 16 = 8.

Step 3: Now let us bring down 09 which is the new dividend. Add the old divisor with the same number 4 + 4 = 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 ≤ 809. Let us consider n as 9, now 89 x 9 = 801.

Step 6: Subtract 809 from 801; the difference is 8, and the quotient is 49.

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 800.

Step 8: Now we need to find the new divisor that is 981 because 981 × 8 = 7848.

Step 9: Subtracting 7848 from 8000, we get the result 152.

Step 10: Now the quotient is 49.08.

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 √2409 is approximately 49.08.

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 2409 using the approximation method.

Step 1: Now we have to find the closest perfect square of √2409. The smallest perfect square less than 2409 is 2304, and the largest perfect square greater than 2409 is 2500. √2409 falls somewhere between 48 and 50.

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 (2409 - 2304) ÷ (2500 - 2304) = 0.525

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 48 + 0.525 = 48.525, so the square root of 2409 is approximately 49.085.

• Square root: The square root of a number is the value that, when multiplied by itself, gives the original number. For example, √16 = 4.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction; it has a non-repeating, non-terminating decimal expansion.

• Radical: A radical is a symbol (√) that represents the root of a number.

• Perfect square: A perfect square is a number that is the square of an integer, such as 1, 4, 9, 16, etc.

• Exponential form: In mathematics, exponential form expresses a number using a base and an exponent, such as (2409)^(1/2) for the square root of 2409.