Square Root of 386

The square root is the inverse of the square of the number. 386 is not a perfect square. The square root of 386 is expressed in both radical and exponential form. In the radical form, it is expressed as √386, whereas (386)^(1/2) in exponential form. √386 ≈ 19.64688, 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 like 386, methods such as 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 386 is broken down into its prime factors:

Step 1: Finding the prime factors of 386 Breaking it down, we get 2 x 193, where 193 is a prime number.

Step 2: Now we found out the prime factors of 386. The second step is to make pairs of those prime factors. Since 386 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 386 using prime factorization is not feasible.

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 386, we need to group it as 86 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 lesser than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.

Step 3: Now let us bring down 86, making the new dividend 286. Add the old divisor with the same number, 1 + 1, giving us 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, and we need to find the value of n.

Step 5: Finding 2n × n ≤ 286, let us consider n as 9, then 2 x 9 x 9 = 261.

Step 6: Subtract 261 from 286, the difference is 25, and the quotient becomes 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. The new dividend is 2500.

Step 8: Find the new divisor, which is 193 because 1939 x 9 = 17451.

Step 9: Subtracting 17451 from 25000, we get the result 7549.

Step 10: Now the quotient is approximately 19.6.

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 √386 is approximately 19.65.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us learn how to find the square root of 386 using the approximation method.

Step 1: Find the closest perfect square of √386.

The smallest perfect square less than 386 is 361, and the largest perfect square greater than 386 is 400.

√386 falls somewhere between 19 and 20.

Step 2: Apply the formula:

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

Using the formula, (386 - 361) ÷ (400 - 361) = 25 ÷ 39 ≈ 0.641.

Adding the initial integer value to the decimal, 19 + 0.641 = 19.641.

Thus, the square root of 386 is approximately 19.641.

• Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, which 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, the positive square root, known as the principal square root, is typically used due to its practical applications.
   
• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
   
• Long division method: A method used to divide larger numbers by breaking them down into a series of easier steps. It is often used to find square roots of non-perfect squares.