Square Root of 246

The square root is the inverse of the square of the number. 246 is not a perfect square. The square root of 246 is expressed in both radical and exponential form. In the radical form, it is expressed as √246, whereas 246^(1/2) in the exponential form. √246 ≈ 15.68439, 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 246 is broken down into its prime factors.

Step 1: Finding the prime factors of 246 Breaking it down, we get 2 × 3 × 41: 2^1 × 3^1 × 41^1

Step 2: Now we found out the prime factors of 246. Since 246 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating √246 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 246, we need to group it as 46 and 2.

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

Step 3: Now let us bring down 46, 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 2n, where n is a digit we need to find such that 2n × n ≤ 146.

Step 5: Let n be 7, so 27 × 7 = 189, which is greater than 146, so we try n as 5.

Step 6: 25 × 5 = 125, now subtract 125 from 146, the difference is 21.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down two zeroes, making the new dividend 2100.

Step 8: Now we need to find the new divisor, which will be 2 times the quotient plus a digit. So, it is 31, and we find 315 × 5 = 1575.

Step 9: Subtracting 1575 from 2100, we get the result 525.

Step 10: The quotient is now 15.6.

Step 11: Continue doing these steps until we get two numbers after the decimal point. So the square root of √246 is approximately 15.68.

The approximation method is another method for finding the 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 246 using the approximation method.

Step 1: Now we have to find the closest perfect square roots to √246. The smallest perfect square less than 246 is 225, and the largest perfect square greater than 246 is 256. √246 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). Using the formula: (246 - 225) ÷ (256 - 225) = 21 ÷ 31 ≈ 0.677 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 + 0.677 = 15.677.

Therefore, the square root of 246 is approximately 15.68.

• 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 the principal square root.
   
• Prime factorization: The process of expressing a number as the product of its prime numbers.
   
• Approximation: The method of finding a value that is close enough to the correct answer, usually with some degree of tolerance or uncertainty.