Square Root of 328

The square root is the inverse of squaring a number. 328 is not a perfect square. The square root of 328 can be expressed in both radical and exponential forms. In radical form, it is expressed as √328, whereas in exponential form it is (328)^(1/2). √328 ≈ 18.11077, which is an irrational number because it cannot be expressed as a fraction 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 squares like 328, the long division and approximation methods are commonly used. Let us now explore these methods:

• Prime factorization method
• Long division method
• Approximation method

The prime factorization of a number is the product of its prime factors. Let's see how 328 can be broken down into its prime factors:

Step 1: Finding the prime factors of 328 Breaking it down, we get 2 x 2 x 2 x 41: 2^3 x 41

Step 2: Since 328 is not a perfect square, we cannot pair all the prime factors.

Therefore, calculating √328 using prime factorization directly is not feasible.

The long division method is effective for non-perfect squares. Here's how to find the square root using this method, step by step:

Step 1: Group the digits of 328 from right to left as 28 and 3.

Step 2: Find the largest number whose square is less than or equal to 3. This number is 1 since 1 x 1 = 1. Subtract 1 from 3 to get a remainder of 2.

Step 3: Bring down the next group of digits, 28, to get a new dividend of 228.

Step 4: Double the quotient (1) to get 2, which becomes the new divisor prefix.

Step 5: Find a digit n such that 2n x n is less than or equal to 228. Here, n is 8, since 28 x 8 = 224.

Step 6: Subtract 224 from 228 to get a remainder of 4. The quotient is now 18.

Step 7: Add a decimal point to the quotient and bring down two zeros to make the new dividend 400.

Step 8: Double the current quotient (18) to get 36, and find n such that 36n x n is less than or equal to 400. Here, n is 1, since 361 x 1 = 361.

Step 9: Subtract 361 from 400 to get a remainder of 39.

Step 10: The quotient is now 18.1. Continue the process to get more decimal places as needed.

The approximate value of √328 is 18.11.

The approximation method is another way to find square roots. Here's how to use it for 328:

Step 1: Identify the closest perfect squares around √328.

The closest perfect squares are 324 (18^2) and 361 (19^2). Thus, √328 is between 18 and 19.

Step 2: Use the formula (Given number - lower perfect square) / (higher perfect square - lower perfect square) to find the decimal. (328 - 324) / (361 - 324) = 4 / 37 ≈ 0.108 Adding this to 18 gives 18 + 0.108 ≈ 18.108.

Therefore, the square root of 328 is approximately 18.11.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4^2 = 16.

• Irrational number: An irrational number cannot be expressed as a simple fraction. Examples include √2 and π.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 16 is a perfect square because it is 4^2.

• Long division method: A technique used to find the square root of non-perfect squares through a step-by-step division process.

• Approximation method: A method used to estimate the square root of a number by identifying nearby perfect squares and using linear interpolation.