Square Root of 393

The square root is the inverse of the square of a number. 393 is not a perfect square. The square root of 393 is expressed in both radical and exponential form. In the radical form, it is expressed as √393, whereas in exponential form it is (393)^(1/2). √393 ≈ 19.823, 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 393 is broken down into its prime factors.

Step 1: Finding the prime factors of 393 Breaking it down, we get 3 × 131: 3^1 × 131^1

Step 2: Now that we found the prime factors of 393, the second step is to make pairs of those prime factors. Since 393 is not a perfect square, the digits of the number can’t be grouped into pairs.

Therefore, calculating √393 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 393, we need to group it as 93 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 × 1 is less than or equal to 3. Now the quotient is 1, and after subtracting 1, the remainder is 2.

Step 3: Now let us bring down 93, making the new dividend 293. Add the old divisor with the same number, 1 + 1, which gives us 2 as the new divisor.

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

Step 5: The next step is finding 2n × n ≤ 293. Let us consider n as 9, now 29 × 9 = 261.

Step 6: Subtract 261 from 293, and the difference is 32. 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 3200.

Step 8: Now we need to find the new divisor, which is 398, because 398 × 8 = 3184.

Step 9: Subtracting 3184 from 3200, we get the result 16.

Step 10: Now the quotient is 19.8.

Step 11: Continue doing these steps until we get two numbers after the decimal point.

So the square root of √393 ≈ 19.82.

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 393 using the approximation method.

Step 1: Now we have to find the closest perfect square of √393.

The smallest perfect square less than 393 is 361, and the largest perfect square greater than 393 is 400.

√393 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)

Using the formula: (393 - 361) ÷ (400 - 361) = 32 ÷ 39 ≈ 0.82

Adding the decimal value to the lower perfect square root: 19 + 0.82 = 19.82, so the square root of 393 is approximately 19.82.

• Square root: A square root is the inverse of a square. Example: 4² = 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 usually the positive square root that is used in real-world applications. This is known as the principal square root.
   
• Non-perfect square: A non-perfect square is a number that does not have an integer as its square root.
   
• Radical form: Radical form expresses the square root of a number using the symbol √.