Square Root of 388

The square root is the inverse of the square of a number. 388 is not a perfect square. The square root of 388 is expressed in both radical and exponential form. In the radical form, it is expressed as √388, whereas in exponential form, it is expressed as (388)^(1/2). √388 ≈ 19.69772, 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 388 is broken down into its prime factors:

Step 1: Finding the prime factors of 388 Breaking it down, we get 2 x 2 x 97.

Step 2: Now that we have found the prime factors of 388, the second step is to make pairs of those prime factors. Since 388 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 388 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. This method involves checking 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: Group the numbers from right to left. In the case of 388, group it as 88 and 3.

Step 2: Find n whose square is 3 or less. 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 x 1 from 3, and the remainder is 2.

Step 3: Bring down 88 to make the new dividend 288. Add the old divisor with the same number: 1 + 1 equals 2, which will be our new divisor.

Step 4: The new divisor will be 2n. Find the value of n such that 2n x n ≤ 288. Let n be 9, then 29 x 9 = 261.

Step 5: Subtract 261 from 288; the difference is 27, and the quotient is 19.

Step 6: Add a decimal point and two zeroes to the dividend. The new dividend is 2700.

Step 7: Find the new divisor that is 393, since 393 x 7 = 2751, which is too large. Instead, use 392 x 6 = 2352.

Step 8: Subtract 2352 from 2700 to get 348. The new quotient is 19.6.

Step 9: Continue these steps until the desired accuracy is reached.

So the square root of √388 ≈ 19.70.

The approximation method is another method for finding square roots. It's an easy method to find the square root of a given number. Now let us learn how to find the square root of 388 using the approximation method.

Step 1: Find the closest perfect squares to √388.

The smallest perfect square less than 388 is 361, and the largest perfect square less than 388 is 400.

√388 falls between 19 and 20.

Step 2: Apply the formula

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

Using the formula (388 - 361) / (400 - 361) = 27 / 39 ≈ 0.6923.

Add the decimal to the smaller perfect square root: 19 + 0.6923 ≈ 19.6923.

Thus, the square root of 388 is approximately 19.6923.

• 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 cannot be written 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, it is usually the positive square root that is more prominent, known as the principal square root.
   
• Prime factorization: Breaking down a number into its basic prime number components. Example: The prime factorization of 388 is 2 x 2 x 97.
   
• Long division method: A technique used to find the square roots of non-perfect squares, involving a step-by-step division process to approximate the root.