Square Root of 3925

The square root is the inverse of the square of the number. 3925 is not a perfect square. The square root of 3925 is expressed in both radical and exponential form. In the radical form, it is expressed as √3925, whereas (3925)^(1/2) in the exponential form. √3925 ≈ 62.641839, 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 3925 is broken down into its prime factors:

Step 1: Finding the prime factors of 3925 Breaking it down, we get 5 x 5 x 157: 5^2 x 157^1

Step 2: Now we found out the prime factors of 3925. The second step is to make pairs of those prime factors. Since 3925 is not a perfect square, the digits of the number can’t be grouped in pairs completely.

Therefore, calculating 3925 using prime factorization alone is insufficient.

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 3925, we need to group it as 39 and 25.

Step 2: Now we need to find n whose square is less than or equal to 39. We can say n as ‘6’ because 6 x 6 = 36, which is less than or equal to 39. Now the quotient is 6, and after subtracting 36 from 39, the remainder is 3.

Step 3: Now let us bring down 25, which is the new dividend. Add the old divisor with the same number 6 + 6, we get 12, which will be our new divisor.

Step 4: The new divisor will be 12n, where n is to be determined such that 12n x n ≤ 325.

Step 5: Let n = 2, so 122 x 2 = 244, which is less than 325.

Step 6: Subtract 244 from 325, the difference is 81, and the quotient becomes 62.

Step 7: 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 8100.

Step 8: Now we need to find the new divisor, which is 124 because 1244 x 4 = 4976.

Step 9: Subtract 4976 from 8100, we get the result 3124.

Step 10: The quotient is 62.6

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero.

So the square root of √3925 is approximately 62.64.

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 3925 using the approximation method.

Step 1: Now we have to find the closest perfect square of √3925. The smallest perfect square less than 3925 is 3844 (62^2), and the largest perfect square greater than 3925 is 4096 (64^2). √3925 falls somewhere between 62 and 63.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). (3925 - 3844) / (4096 - 3844) = 81 / 252 = 0.32 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 62 + 0.32 = 62.32.

So the square root of 3925 is approximately 62.32 in this approximation.

• 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 a positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as a principal square root.
   
• Prime factorization: A method of expressing a number as the product of its prime factors.
   
• Long division method: A procedure used for dividing a number to determine its square root, particularly for non-perfect squares.