Square Root of 239

The square root is the inverse of the square of the number. 239 is not a perfect square. The square root of 239 is expressed in both radical and exponential form. In the radical form, it is expressed as √239, whereas (239)^(1/2) in the exponential form. √239 ≈ 15.459, 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 long-division and approximation methods 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 239 is broken down into its prime factors.

Step 1: Finding the prime factors of 239 239 is a prime number, so it cannot be broken down further into smaller prime numbers.

Step 2: Since 239 is a prime number, the prime factorization method cannot be used to find its square root directly. Therefore, calculating √239 using prime factorization is not applicable.

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 239, we group it as 39 and 2.

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

Step 3: Now let us bring down 39, making the new dividend 139. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.

Step 4: The new divisor will be 2n, and we need to find the value of n such that 2n × n ≤ 139. Let us consider n as 6, now 26 × 6 = 156, which is greater than 139, so try n as 5, 25 × 5 = 125.

Step 5: Subtract 125 from 139; the difference is 14, and the quotient is 15.

Step 6: 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 1400.

Step 7: Now we need to find the new divisor that is 310 because 310 × 4 = 1240.

Step 8: Subtracting 1240 from 1400, we get 160.

Step 9: Now the quotient is 15.4.

Step 10: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.

So the square root of √239 ≈ 15.459.

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 239 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √239. The smallest perfect square less than 239 is 225, and the largest perfect square greater than 239 is 256. √239 falls somewhere between 15 and 16.

Step 2: Now we need to apply the interpolation formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) (239 - 225) ÷ (256 - 225) = 14 ÷ 31 ≈ 0.4516 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 + 0.4516 ≈ 15.4516.

So the square root of 239 is approximately 15.459.

• 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.
   
• Prime number: A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. Example: 239.
   
• Radical form: A way to express a number using a radical symbol, for example, √239.
   
• Decimal: A number that includes a fraction represented in a system based on powers of 10, for example, 15.459.