Square Root of 5328

The square root is the inverse of the square of a number. 5328 is not a perfect square. The square root of 5328 is expressed in both radical and exponential forms. In the radical form, it is expressed as √5328, whereas in exponential form, it is expressed as (5328)^(1/2). √5328 ≈ 73, which is an irrational number because it cannot be expressed as a ratio of two integers.

The prime factorization method is typically used for perfect square numbers. However, for non-perfect square numbers, the long division method and approximation method are more suitable. 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 5328 is broken down into its prime factors.

Step 1: Finding the prime factors of 5328 Breaking it down, we get 2 x 2 x 2 x 2 x 3 x 3 x 37: 2^4 x 3^2 x 37

Step 2: Now that we have found the prime factors of 5328, the second step is to make pairs of those prime factors. Since 5328 is not a perfect square, the digits of the number can’t be grouped into pairs completely.

Therefore, calculating 5328 using prime factorization is not straightforward.

The long division method is particularly used 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 numbers from right to left. In the case of 5328, we need to group it as 28 and 53.

Step 2: Now we need to find a number whose square is closest to 53. We can say n is '7' because 7 x 7 = 49, which is less than 53. Now the quotient is 7, and after subtracting 49 from 53, the remainder is 4.

Step 3: Bring down 28, which is the new dividend. Add the old divisor with the same number: 7 + 7 = 14, which will be our new divisor.

Step 4: The new divisor is 2n. Now we need to find the value of n.

Step 5: The next step is finding 14n x n ≤ 428. Let's consider n as 3, now 143 x 3 = 429, which is greater than 428, so we try n = 2, then 142 x 2 = 284.

Step 6: Subtract 284 from 428; the difference is 144.

Step 7: Since the dividend is not zero, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 14400.

Step 8: Now we need to find the new divisor, which is 1472 because 1472 x 9 = 13248.

Step 9: Subtract 13248 from 14400; we get the result 1152.

Step 10: Now the quotient is 73.9

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √5328 is approximately 73.

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 5328 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √5328.

The smallest perfect square less than 5328 is 5041 (71^2), and the largest perfect square greater than 5328 is 5184 (72^2). √5328 falls somewhere between 72 and 73.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Using the formula (5328 - 5041) / (5184 - 5041) ≈ 0.71. Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 72 + 0.71 ≈ 72.71, so the square root of 5328 is approximately 72.71.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the inverse of squaring is finding the square root, so √16 = 4.

• Irrational number: An irrational number is a number that cannot be written as a simple fraction, i.e., it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

• Principal square root: A number has both positive and negative square roots; however, we usually refer to the positive square root due to its utility in real-world applications. This is known as the principal square root.

• Long division method: A step-by-step method to find the square root of a number, particularly useful for non-perfect squares.

• Approximation method: A method used to estimate the square root of a number by finding the closest perfect squares above and below the given number and approximating its value.