Square Root of 368

The square root is the inverse of the square of the number. 368 is not a perfect square. The square root of 368 is expressed in both radical and exponential form. In the radical form, it is expressed as √368, whereas (368)^(1/2) in the exponential form. √368 ≈ 19.1833, 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 368 is broken down into its prime factors:

Step 1: Finding the prime factors of 368 Breaking it down, we get 2 x 2 x 2 x 2 x 23: 2^4 x 23^1

Step 2: Now we found out the prime factors of 368. The second step is to make pairs of those prime factors. Since 368 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 368 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 368, we need to group it as 68 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 = 1 is less than or equal to 3. Now the quotient is 1, after subtracting 1, the remainder is 2.

Step 3: Now let us bring down 68, 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 will be 20 (2n), and we need to find a number n such that 20n x n ≤ 268.

Step 5: Consider n as 9, then 209 x 9 = 1881, which is less than 2680, so n = 9.

Step 6: Subtract 1881 from 2680, the difference is 799, 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 79900.

Step 8: Now we need to find the new divisor that is 193 because 1939 x 9 = 17451.

Step 9: Subtracting 17451 from 79900, we get the result 62449.

Step 10: Now the quotient is 19.1

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 √368 ≈ 19.18

The approximation method is another method for finding square roots, and 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 368 using the approximation method.

Step 1: Find the closest perfect squares of √368.

The smallest perfect square less than 368 is 361 (19^2), and the largest perfect square greater than 368 is 400 (20^2).

√368 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: (368 - 361) / (400 - 361) = 7 / 39 ≈ 0.179

Using the formula, we identify the decimal point of our square root.

The next step is adding the value we got initially to the decimal number, which is 19 + 0.179 = 19.179, so the square root of 368 is approximately 19.179.

• Square root: A square root is the inverse of a square. Example: 5^2 = 25, and the inverse of the square is the square root; that is, √25 = 5.
   
• 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.
   
• Prime factorization: The expression of a number as the product of its prime factors. For example, the prime factorization of 368 is 2^4 × 23.
   
• Long division method: A method used to find the square root of non-perfect squares by dividing the number into groups and iteratively finding the quotient.