Square Root of 9360

The square root is the inverse of squaring a number. 9360 is not a perfect square. The square root of 9360 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √9360, whereas in exponential form it is (9360)^(1/2). √9360 ≈ 96.79, 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 like 9360, the long-division method and approximation method are more appropriate. Let us explore these methods: 

• Prime factorization method
• Long division method
• Approximation method

Prime factorization involves breaking down a number into its prime factors. Let us examine the prime factorization of 9360.

Step 1: Finding the prime factors of 9360

Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 5 x 13 x 3: 2^4 x 3^2 x 5 x 13

Step 2: We have found the prime factors of 9360. The next step is to pair these prime factors. Since 9360 is not a perfect square, these factors can't be perfectly paired, making the calculation of √9360 by prime factorization alone impossible.

The long division method is especially useful for non-perfect square numbers. This method involves finding the closest perfect square number to the given number. Let's find the square root using the long division method, step by step.

Step 1: Group the digits of 9360 from right to left as 60 and 93.

Step 2: Find a number (n) whose square is closest to 93. Here, n is 9 because 9 x 9 = 81, which is less than 93. Subtract 81 from 93 to get a remainder of 12.

Step 3: Bring down the next pair of digits, 60, to get 1260 as the new dividend. Double the quotient (9), which gives 18, forming the new divisor.

Step 4: Find the largest digit (n) such that 18n x n is less than or equal to 1260. The appropriate n is 6, as 186 x 6 = 1116.

Step 5: Subtract 1116 from 1260 to get 144.

Step 6: Bring down another pair of zeros to make it 14400.

Step 7: Double the previous divisor (186) and add the new digit (6) to get 192 as the new divisor base.

Step 8: Find the largest digit (n) such that 192n x n is less than or equal to 14400.

Continue this process to get an approximate value of √9360 ≈ 96.79.

The approximation method is another approach to finding square roots, offering a simpler calculation for √9360.

Step 1: Identify the closest perfect squares around 9360. The closest perfect square below 9360 is 9216 (96^2), and above is 9409 (97^2). √9360 lies between 96 and 97.

Step 2: Use the approximation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (9360 - 9216) / (9409 - 9216) ≈ 0.79 Add this to the integer part (96) to approximate √9360 as 96 + 0.79 = 96.79.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 10^2 = 100, so √100 = 10.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction, with examples including √9360, which is not a perfect square.
   
• Principal square root: The positive square root of a number, often used in practical applications.
   
• Prime factorization: Breaking down a number into its prime factors. For instance, the prime factorization of 9360 is 2^4 x 3^2 x 5 x 13.
   
• Approximation: A method to estimate the square root of numbers that are not perfect squares, such as √9360 ≈ 96.79.