Square Root of 3842

The square root is the inverse of the square of a number. 3842 is not a perfect square. The square root of 3842 is expressed in both radical and exponential form. In the radical form, it is expressed as √3842, whereas (3842)^(1/2) in the exponential form. √3842 ≈ 61.9935, 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 3842 is broken down into its prime factors.

Step 1: Finding the prime factors of 3842 Breaking it down, we find 3842 = 2 x 1921, and 1921 is 13 x 13 x 11.

Step 2: Now we found out the prime factors of 3842. Since 3842 is not a perfect square, the digits of the number can't be grouped in pairs.

Therefore, calculating 3842 using prime factorization is not straightforward.

The long division method is particularly useful 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 3842, we group it as 38 and 42.

Step 2: Now we need to find n whose square is less than or equal to 38. We can say n as ‘6’ because 6 x 6 = 36, which is less than 38. Now the quotient is 6, and after subtracting 36 from 38, the remainder is 2.

Step 3: Now let us bring down 42, making the new dividend 242. Add the old divisor with the same number: 6 + 6 = 12, which will be our new divisor.

Step 4: The new divisor will be 12n. We need to find the value of n.

Step 5: Finding 12n × n ≤ 242, let's consider n as 2, now 12 x 2 = 24.

Step 6: Subtract 24 from 242, the difference is 218, and the quotient is 62.

Step 7: Since the dividend is larger than the new divisor, we bring down two additional zeros (as decimal points) to continue the process.

Step 8: Continue the process to find the decimal places.

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Let us now learn how to find the square root of 3842 using the approximation method.

Step 1: Now, we have to find the closest perfect square of √3842. The smallest perfect square less than 3842 is 3721 (61^2), and the largest perfect square greater than 3842 is 3969 (63^2). √3842 falls somewhere between 61 and 63.

Step 2: Now we apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Going by the formula (3842 - 3721) / (3969 - 3721) ≈ 0.9935 Adding this to the lower bound: 61 + 0.9935 ≈ 61.9935.

So, the square root of 3842 is approximately 61.9935.

• 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 a product of its prime factors.
   
• Approximation method: A method used to find an approximate value of a number when an exact value is not needed or is difficult to obtain.