Square Root of 392

The square root is the inverse of the square of the number. 392 is not a perfect square. The square root of 392 is expressed in both radical and exponential form. In the radical form, it is expressed as √392, whereas 392^(1/2) in the exponential form. √392 ≈ 19.79899, 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 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 392 is broken down into its prime factors.

Step 1: Finding the prime factors of 392 Breaking it down, we get 2 x 2 x 2 x 7 x 7: 2^3 x 7^2

Step 2: Now we found out the prime factors of 392. The next step is to make pairs of those prime factors. Since 392 is not a perfect square, therefore the digits of the number can’t be grouped in a complete pair.

Therefore, calculating √392 using prime factorization directly as a perfect square is not possible.

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 392, we need to group it as 92 and 3.

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

Step 3: Now let us bring down 92, which is 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, 20n, needs to be determined by finding n.

Step 5: The next step is finding 20n × n ≤ 292. Let us consider n as 1, now 20 x 1 x 1 = 20.

Step 6: Subtract 20 from 292; the difference is 272, 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 27200.

Step 8: Now we need to find the new divisor, 398, because 398 x 7 = 2786.

Step 9: Subtracting 2786 from 27200, we get the result 442.

Step 10: Now the quotient is 19.7.

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

So, the square root of √392 ≈ 19.80

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 392 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √392.

The smallest perfect square less than 392 is 361, and the largest perfect square greater than 392 is 400.

√392 falls somewhere between 19 and 20.

Step 2: Now we need to apply the formula:

(Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).

Going by the formula (392 - 361) ÷ (400 - 361) = 31 / 39 ≈ 0.7949

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 + 0.7949 ≈ 19.80, so the square root of 392 is approximately 19.80.

• 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, 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.
   
• Prime Factorization: Breaking down a composite number into a product of its prime factors.
   
• Long Division Method: A method used to find the square root of non-perfect squares by dividing and estimating iteratively.