Square Root of 396

The square root is the inverse of the square of a number. 396 is not a perfect square. The square root of 396 is expressed in both radical and exponential form. In the radical form, it is expressed as √396, whereas (396)^(1/2) in exponential form. √396 ≈ 19.8997, 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, the prime factorization method is not used for non-perfect square numbers where 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 396 is broken down into its prime factors:

Step 1: Finding the prime factors of 396 Breaking it down, we get 2 x 2 x 3 x 3 x 11: 2^2 x 3^2 x 11

Step 2: Now we found out the prime factors of 396. The second step is to make pairs of those prime factors. Since 396 is not a perfect square, the digits of the number can’t be grouped entirely into pairs.

Therefore, calculating the square root of 396 using prime factorization results in an approximate value.

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 396, we need to group it as 96 and 3.

Step 2: Now we need to find n whose square is less than or equal to 3. We can say n as ‘1’ because 1 x 1 is equal to 1. Now the quotient is 1 and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 96, which becomes the new dividend. Add the old divisor with the same number, 1 + 1, we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 296. Let us consider n as 9, now 29 x 9 = 261.

Step 6: Subtract 261 from 296, the difference is 35, and the quotient is 19.

Step 7: Since the dividend is less than the divisor, 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 3500.

Step 8: Now we need to find the new divisor that is 399 because 399 x 9 = 3591.

Step 9: Subtracting 3591 from 3500, we get the result 91.

Step 10: Now the quotient is 19.8.

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 √396 is approximately 19.90.

The approximation method is another method for finding 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 396 using the approximation method.

Step 1: Now we have to find the closest perfect square of √396.

The smallest perfect square less than 396 is 361 (19^2), and the largest perfect square greater than 396 is 400 (20^2).

√396 falls somewhere between 19 and 20.

Step 2: Now we need to apply the formula:

(Given number - smaller perfect square) / (larger perfect square - smaller perfect square)

Using the formula: (396 - 361) / (400 - 361) = 35 / 39 ≈ 0.90

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 19 + 0.90 = 19.90.

So the square root of 396 is approximately 19.90.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root that is √16 = 4.
   
• 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, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Perfect square: A perfect square is a number that can be expressed as the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example, 7.86, 8.65, and 9.42 are decimals.