Square Root of 604

The square root is the inverse operation of squaring a number. 604 is not a perfect square. The square root of 604 can be expressed in both radical and exponential forms. In radical form, it is expressed as √604, whereas in exponential form, it is (604)^(1/2). √604 ≈ 24.5764, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method is typically used for perfect square numbers. For non-perfect square numbers like 604, the long division method and approximation method are more suitable. Let's explore these 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 604 is broken down into its prime factors:

Step 1: Finding the prime factors of 604 Breaking it down, we get 2 x 2 x 151: 2^2 x 151

Step 2: We found the prime factors of 604. The next step is to make pairs of those prime factors. Since 604 is not a perfect square, the digits of the number cannot be grouped into pairs. Therefore, calculating √604 using prime factorization is not possible.

The long division method is particularly useful 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 604, we group it as 04 and 6.

Step 2: Find n whose square is less than or equal to 6. We can select n as 2 because 2 x 2 = 4, which is less than 6. The quotient is 2, and the remainder is 6 - 4 = 2.

Step 3: Bring down the next pair, which is 04. Now the new dividend is 204. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be 4n. We need to find n such that 4n x n ≤ 204. Let's consider n as 5; then, 45 x 5 = 225, which is greater than 204. So, we try n as 4; then, 44 x 4 = 176.

Step 5: Subtract 176 from 204. The difference is 28, and the quotient is 24.

Step 6: Since the dividend is less than the divisor, add a decimal point. Add two zeroes to the dividend, making it 2800.

Step 7: Find the new divisor: 488 x 6 = 2928, which is more than 2800. Try 487 x 5 = 2435.

Step 8: Subtract 2435 from 2800, getting 365. Bring down two more zeroes, making it 36500.

Step 9: Continue doing these steps until we get the desired number of decimal places. So, √604 is approximately 24.576.

The approximation method is another way to find square roots and is an easy method to find the square root of a number. Let's learn how to find the square root of 604 using the approximation method.

Step 1: Find the closest perfect squares of √604. The smallest perfect square less than 604 is 576, and the largest perfect square greater than 604 is 625. √604 lies between 24 and 25.

Step 2: Apply the interpolation formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square) (604 - 576) ÷ (625 - 576) = 28 ÷ 49 ≈ 0.571 Using this method, we identified the decimal portion of our square root. Adding this to the whole number, we get 24 + 0.571 = 24.571. So the square root of 604 is approximately 24.571.

• Square root: A square root is the inverse operation of squaring a number. For example, 5^2 = 25, and the inverse operation is √25 = 5.

• Irrational number: An irrational number cannot be expressed as a simple fraction or ratio of two integers. For example, √2 is an irrational number.

• 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.

• Prime factorization: The expression of a number as a product of its prime factors. For example, the prime factorization of 28 is 2 x 2 x 7.

• Long division method: A method used to calculate the square root of a number, especially for non-perfect squares, involving dividing and averaging processes.