Square Root of 42

The square root is the inverse of the square of the number. 42 is not a perfect square. The square root of 42 is expressed in both radical and exponential form.

In the radical form, it is expressed as √42, whereas (42)^(1/2) in the exponential form. √42 ≈ 6.4807, 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 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 42 is broken down into its prime factors.

Step 1: Finding the prime factors of 42 Breaking it down, we get 2 x 3 x 7.

Step 2: Now we found the prime factors of 42. The second step is to make pairs of those prime factors. Since 42 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 42 using prime factorization is not exact.

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 42, it is already a two-digit number, so we can consider it as 42.

Step 2: Now we need to find a number whose square is less than or equal to 42. We can say n as '6' because 6 x 6 = 36, which is less than 42. Now, the quotient is 6, and after subtracting 36 from 42, the remainder is 6.

Step 3: Add a decimal point to the quotient and bring down two zeros, making the new dividend 600.

Step 4: Double the current quotient to get 12, which will be part of our new divisor.

Step 5: Find a digit, let's say m, such that 12m x m is less than or equal to 600. Here, m is 4, since 124 x 4 = 496.

Step 6: Subtract 496 from 600, the difference is 104, and the quotient is 6.4.

Step 7: Continue this process to find more decimal places if needed.

So, the square root of √42 is approximately 6.48.

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 42 using the approximation method.

Step 1: Now we have to find the closest perfect square of √42. The smallest perfect square less than 42 is 36 and the largest perfect square greater than 42 is 49. √42 falls somewhere between 6 and 7.

Step 2: Now we need to apply the formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square) (42 - 36) / (49 - 36) = 6/13 ≈ 0.46

Using the formula, we determined the decimal part of our square root. The next step is adding the value we got initially to the decimal number: 6 + 0.46 ≈ 6.46.

Thus, the square root of 42 is approximately 6.46.

• Square root: A square root is the inverse of a square. For 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, but the positive square root is typically used due to its applications in the real world.
   
• Perfect square: A perfect square is a number that is the square of an integer. For example: 36 is a perfect square because it is 6^2.
   
• Long division method: A mathematical procedure used to find the square root of non-perfect square numbers by dividing the number into pairs of digits.