Square Root of 631

The square root is the inverse of the square of the number. 631 is not a perfect square. The square root of 631 is expressed in both radical and exponential form. In the radical form, it is expressed as √631, whereas (631)^(1/2) in the exponential form. √631 ≈ 25.127, 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.

For non-perfect square numbers, methods like the long division method and approximation method are used instead of the prime factorization method. Let us now learn the following methods:

• Long division method
• Approximation method

Since 631 is not a perfect square, we cannot use the prime factorization method effectively to find its square root. However, for completeness, here is its prime factorization:

Step 1: Finding the prime factors of 631 Breaking it down, we get 631 = 17 x 37.

Step 2: Since 631 is not a perfect square, the digits cannot be grouped in pairs. Therefore, calculating 631 using prime factorization does not yield an exact square root.

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 631, we need to group it as 31 and 6.

Step 2: Now we need to find n whose square is 6. We can say n as ‘2’ because 2 x 2 = 4 is less than or equal to 6. Now the quotient is 2; after subtracting 4 from 6, the remainder is 2.

Step 3: Now let us bring down 31, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 4n as the new divisor, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 231. Let us consider n as 5, now 45 x 5 = 225.

Step 6: Subtract 225 from 231; the difference is 6, and the quotient is 25.

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 600.

Step 8: Now we need to find the new divisor, which is 501. We consider n as 1, because 501 x 1 = 501.

Step 9: Subtracting 501 from 600, we get the result 99.

Step 10: Now the quotient is 25.1.

Step 11: Continue doing these steps until we get two decimal places. Suppose if there is no decimal value, continue till the remainder is zero. So the square root of √631 ≈ 25.13.

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 631 using the approximation method.

Step 1: Now we have to find the closest perfect squares of √631. The smallest perfect square below 631 is 625, and the largest perfect square above 631 is 676. √631 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (631 - 625) / (676 - 625) = 6 / 51 ≈ 0.118. Adding the value we got initially to the decimal number, which is 25 + 0.118 ≈ 25.118, so the square root of 631 is approximately 25.118.

• Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 5² = 25, and the inverse of the square is the square root, so √25 = 5.
   
• Irrational number: An irrational number 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, the positive square root is more commonly used due to its real-world applications. This is known as the principal square root.
   
• Perfect square: A perfect square is a number that is the square of an integer. Example: 16 is a perfect square because 4² = 16.
   
• Long division method: A technique used to find the square root of non-perfect squares by breaking the number into manageable parts and solving iteratively.