Square Root of 9600

The square root is the inverse of the square of a number. 9600 is not a perfect square, but it can be simplified significantly. The square root of 9600 is expressed in both radical and exponential form. In radical form, it is expressed as √9600, whereas in exponential form, it is (9600)^(1/2). √9600 = 97.9796, 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 simplifying square roots, especially for perfect square numbers. For non-perfect square numbers, methods like 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 9600 is broken down into its prime factors.

Step 1: Finding the prime factors of 9600

Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 3 x 5 x 5 x 5: 2^5 x 3 x 5^3

Step 2: Now that we have found the prime factors of 9600, we can group them in pairs to simplify the square root. Grouping the pairs, we have (2^5 = 2 x 2 x 2 x 2 x 2) and (5^3 = 5 x 5 x 5). Pairing them, we get (2 x 2) x (5 x 5) x √(2 x 3 x 5). Step 3: Using the pairs, we simplify the square root: √9600 = (2^2 x 5) x √(2 x 3 x 5) = 20√30.

The long division method is particularly useful for non-perfect square numbers. In this method, we should check the closest perfect square numbers surrounding 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 9600, we need to group it as 96 and 00.

Step 2: Now, find n whose square is less than or equal to 96. We can say n as ‘9’ because 9 x 9 = 81 is lesser than 96. Now the quotient is 9, and the remainder is 96 - 81 = 15.

Step 3: Bring down the next pair of digits (00) to the right of the remainder. The new dividend is 1500.

Step 4: Double the current quotient (9), giving us 18, which will be used as part of the new divisor.

Step 5: Find a digit (d) such that 18d x d is less than or equal to 1500. Here, d is determined to be 8, as 188 x 8 = 1504, which is just over 1500.

Step 6: Since 188 x 7 = 1316 is less than 1500, 7 is used. Subtract 1316 from 1500, leaving a remainder of 184.

Step 7: Add a decimal point and two zeros to the remainder, making it 18400. Repeat the process with the new dividend.

Step 8: Continue this process until the desired precision is achieved.

The result of √9600 is approximately 97.9796.

The approximation method is another method for finding square roots; it is an easy way to estimate the square root of a given number. Now let us learn how to find the square root of 9600 using the approximation method.

Step 1: Identify two perfect squares between which 9600 lies. The smallest perfect square less than 9600 is 9216 (96^2), and the largest perfect square more than 9600 is 10000 (100^2).

Step 2: Estimate the square root by interpolation. Use the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (9600 - 9216) ÷ (10000 - 9216) = 384 ÷ 784 ≈ 0.4898

Step 3: Add this decimal to the integer part of the smaller square root: 96 + 0.4898 ≈ 96.49

Step 4: For a more precise answer, refine the approximation using additional methods, resulting in √9600 ≈ 97.9796.

• Square root: A square root is the inverse of squaring a number. For example, 10^2 = 100, and the inverse of the square is the square root, so √100 = 10.
   
• 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.
   
• Principal square root: A number has both positive and negative square roots; however, the positive square root is typically considered because of its practical applications. This is known as the principal square root.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 100 is a perfect square because it is 10^2.
   
• Approximation: Approximation is a method of finding a value or solution close to the exact result, often used when calculating square roots of non-perfect squares.