Square Root of 3328

The square root is the inverse of the square of a number. 3328 is not a perfect square. The square root of 3328 is expressed in both radical and exponential form. In the radical form, it is expressed as √3328, whereas in exponential form it is expressed as (3328)^(1/2). √3328 ≈ 57.693, 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, 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 3328 is broken down into its prime factors:

Step 1: Finding the prime factors of 3328 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 2 x 13: 2^6 x 13

Step 2: Now we have found the prime factors of 3328. The next step is to make pairs of those prime factors. Since 3328 is not a perfect square, the digits of the number can’t be grouped into pairs evenly. Therefore, calculating the square root of 3328 using prime factorization is not straightforward.

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 digits from right to left in pairs. For 3328, we consider 28 and 33.

Step 2: Now, find n whose square is less than or equal to 33. We can take n as 5 because 5^2 = 25, which is less than 33. The quotient is 5, and after subtraction, 33 - 25 = 8.

Step 3: Bring down the next pair of digits, which is 28, to make the new dividend, 828.

Step 4: Add the previous divisor (5) to itself to get 10, and use this as a part of the new divisor.

Step 5: Find a new digit p such that 10p x p is less than or equal to 828. In this case, 107 x 7 = 749.

Step 6: Subtract 749 from 828 to get 79. The quotient is now 57.

Step 7: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend making it 7900.

Step 8: The new divisor is 1149 (1070 + 7 + 7). Find a digit q such that 1149q x q is less than or equal to 7900. Continue this process until you achieve the desired precision. The square root of 3328 is approximately 57.69.

The approximation method is another way to find square roots. It is an easy method to find the square root of a given number. Now, let us learn how to find the square root of 3328 using the approximation method.

Step 1: Identify the nearest perfect squares for 3328. The smallest perfect square less than 3328 is 3249 (57^2) and the largest perfect square more than 3328 is 3364 (58^2). Thus, √3328 falls between 57 and 58.

Step 2: Apply the formula: (Given number - smallest perfect square) / (greater perfect square - smallest perfect square). Using the formula (3328 - 3249) / (3364 - 3249) ≈ 0.69 Add this decimal to the smaller integer value, which is 57. Therefore, 57 + 0.69 = 57.69. The approximate square root of 3328 is 57.69.

• Square root: A square root is the inverse of squaring a number. Example: If 8^2 = 64, then the inverse, or square root, is √64 = 8.

• Irrational number: An irrational number cannot be expressed as a simple fraction p/q, where p and q are integers and q ≠ 0.

• Principal square root: The principal square root of a number is its non-negative square root, often used in real-world applications.

• Prime factorization: The process of breaking down a number into its basic prime factors.

• Long division method: A systematic approach to finding the square root of numbers, especially useful for non-perfect squares.