Square Root of 238

The square root is the inverse of the square of the number. 238 is not a perfect square. The square root of 238 is expressed in both radical and exponential form. In the radical form, it is expressed as √238, whereas (238)^(1/2) in the exponential form. √238 ≈ 15.4272, which is an irrational number because it cannot be expressed in the form 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 238 is broken down into its prime factors.

Step 1: Finding the prime factors of 238 Breaking it down, we get 2 x 7 x 17: 2^1 x 7^1 x 17^1

Step 2: Now we found out the prime factors of 238. The second step is to make pairs of those prime factors. Since 238 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating √238 using prime factorization is impossible.

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 238, we need to group it as 38 and 2.

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

Step 3: Now let us bring down 38, which is the new dividend. Add the old divisor with the same number: 1 + 1, and we get 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: The next step is finding 2n x n ≤ 138. Let us consider n as 5. Now 25 x 5 = 125.

Step 6: Subtract 125 from 138, and the difference is 13, and the quotient is 15.

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 1300.

Step 8: Now we need to find the new digit that makes 305n x n ≤ 1300. The digit is 4 because 3054 x 4 = 12216.

Step 9: Subtracting 12216 from 13000, we get the result 784.

Step 10: Now the quotient is 15.4.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values, continue till the remainder is zero. So the square root of √238 is approximately 15.42.

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 238 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √238. The smallest perfect square less than 238 is 225, and the largest perfect square greater than 238 is 256. √238 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (238 - 225) / (256 - 225) = 13 / 31 ≈ 0.419. 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 15 + 0.419 = 15.419, so the square root of 238 is approximately 15.42.

• 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, 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 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: Prime factorization is the process of expressing a number as the product of its prime factors.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, 16, and 25 are perfect squares.