Square Root of 9604

The square root is the inverse of the square of the number. 9604 is a perfect square. The square root of 9604 is expressed in both radical and exponential form. In the radical form, it is expressed as √9604, whereas (9604)^(1/2) in the exponential form. √9604 = 98, which is a rational number because it can be expressed in the form of p/q, where p and q are integers and q ≠ 0.

The prime factorization method can be used for perfect square numbers. For non-perfect square numbers, the 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 9604 is broken down into its prime factors.

Step 1: Finding the prime factors of 9604

Breaking it down, we get 2 x 2 x 7 x 7 x 7 x 7: 2^2 x 7^4

Step 2: Now we have found the prime factors of 9604. The second step is to make pairs of those prime factors. Since 9604 is a perfect square, the digits of the number can be grouped in pairs. Therefore, the square root of 9604 using prime factorization is 2 x 7 x 7 = 98.

The long division method is particularly useful for finding the square root of numbers, especially when they are not perfect squares. However, let's see how it works for 9604:

Step 1: To begin with, we need to group the numbers from right to left. In the case of 9604, we group it as 96 and 04.

Step 2: Now, find the largest number whose square is less than or equal to 96. We can say that number is 9 because 9 x 9 = 81, which is less than 96. The quotient is 9 and the remainder is 96 - 81 = 15.

Step 3: Bring down the next pair, 04, making the new dividend 1504. Add 9 to itself to get 18 as the new divisor's first part.

Step 4: Guess the largest possible digit to fill the blank in 18_ (let's call it n) such that 18n x n is less than or equal to 1504. After calculating, n comes out to be 8.

Step 5: 188 x 8 = 1504, subtracting 1504 from 1504 gives a remainder of 0.

Thus, the quotient is 98, and the square root of 9604 is 98.

The approximation method can be used to estimate the square roots, but it is more suited for non-perfect squares. For perfect squares like 9604, the exact square root can be found as shown in the previous methods. However, let's see an example of estimation:

Step 1: Identify the closest perfect squares around 9604. We know that 9801 (99^2) and 9604 (98^2) are perfect squares.

Step 2: Since 9604 is exactly 98^2, we don't need further approximation.

The square root of 9604 is precisely 98.

• 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.
   
• Rational number: A rational number is a number that can 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, the positive square root is often used in real-world applications and is known as the principal square root
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• Perfect square: A perfect square is a number that is the square of an integer. For example, 9604 is a perfect square because it is 98^2.
   
• Factorization: The process of breaking down a number into its prime factors. For example, the factorization of 9604 is 2^2 x 7^4.