Square Root of 3924

The square root is the inverse of the square of the number. 3924 is not a perfect square. The square root of 3924 is expressed in both radical and exponential forms. In the radical form, it is expressed as √3924, whereas (3924)^(1/2) in the exponential form. √3924 ≈ 62.62322, 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, for non-perfect square numbers, 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 3924 is broken down into its prime factors.

Step 1: Finding the prime factors of 3924 Breaking it down, we get 2 x 2 x 3 x 3 x 109: 2^2 x 3^2 x 109

Step 2: Now we found out the prime factors of 3924. The second step is to make pairs of those prime factors. Since 3924 is not a perfect square, the digits of the number can’t be grouped in pairs for complete squares.

Therefore, calculating √3924 using prime factorization directly is not possible.

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 3924, we group it as 39 and 24.

Step 2: Now, find the largest number whose square is less than or equal to 39. That number is 6 because 6^2 = 36. The quotient is 6, and the remainder is 3 after subtracting 36 from 39.

Step 3: Bring down the next pair, 24, making the new dividend 324. Double the quotient and place it as the new divisor's base, giving us 12.

Step 4: Find a digit 'n' such that 12n × n is less than or equal to 324. The correct digit is 6, because 126 × 6 = 756, which is more than 324, so try 125 × 5 = 625, which is also more, so correct to 124 × 2 = 248.

Step 5: Subtract 248 from 324, leaving a remainder of 76. The quotient continues to 62.

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, making it 7600.

Step 7: Continue finding the number 'n' for new divisions until you achieve the desired precision.

So the square root of √3924 ≈ 62.62

The approximation method is another technique 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 3924 using the approximation method.

Step 1: Find the closest perfect squares around 3924. The smallest perfect square less than 3924 is 3844 (62^2), and the largest perfect square more than 3924 is 3969 (63^2). √3924 falls somewhere between 62 and 63.

Step 2: Apply the interpolation formula: (Given number - smaller perfect square) / (larger perfect square - smaller perfect square). Using the formula (3924 - 3844) / (3969 - 3844) = 0.64 Now add this decimal to the lower whole number: 62 + 0.64 ≈ 62.64, so the square root of 3924 is approximately 62.64.

• Square root: A square root is the inverse operation of squaring a number. For example, 4^2 = 16, and the square root of 16 is √16 = 4.
   
• Irrational number: An irrational number cannot be expressed as a simple fraction (p/q), where p and q are integers and q ≠ 0.
   
• Perfect square: A perfect square is an integer that is the square of an integer. For example, 36 is a perfect square because it is 6^2.
   
• Long division method: A technique used to find the square root of non-perfect squares by estimating and refining the result through division.
   
• Approximation method: A method of finding an approximate value of the square root of a number by identifying nearby perfect squares and interpolating.