Square Root of 635

The square root is the inverse of the square of the number. 635 is not a perfect square. The square root of 635 is expressed in both radical and exponential form. In the radical form, it is expressed as √635, whereas (635)^(1/2) in the exponential form. √635 ≈ 25.199, 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 635 is broken down into its prime factors.

Step 1: Finding the prime factors of 635 Breaking it down, we get 5 x 127: 5^1 x 127^1

Step 2: Now we found out the prime factors of 635. The second step is to make pairs of those prime factors. Since 635 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 635 using prime factorization is not feasible for finding 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 635, we need to group it as 35 and 6.

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

Step 3: Now let us bring down 35, which is the new dividend. Add the old divisor with the same number 2 + 2 we get 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, we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 235. Let us consider n as 5, now 45 x 5 = 225.

Step 6: Subtract 235 from 225, the difference is 10, 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 1000.

Step 8: Now we need to find the new divisor that is 250 because 2501 x 4 = 1004, which goes beyond 1000, so we choose a lower number.

Step 9: Subtracting 1000 from 999 (using a smaller number for further accuracy), we get the result 1.

Step 10: Now the quotient is 25.1.

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

So the square root of √635 ≈ 25.199

The approximation method is another method for finding the 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 635 using the approximation method.

Step 1: Now we have to find the closest perfect squares to √635. The smallest perfect square less than 635 is 625, and the largest perfect square greater than 635 is 676. √635 falls somewhere between 25 and 26.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (635 - 625) / (676 - 625) = 0.196 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 25 + 0.199 = 25.199, so the square root of 635 is approximately 25.199.

• 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.
   
• Prime factorization: Prime factorization is the method of expressing a number as a product of its prime factors.
   
• Irrational number: An irrational number is a number that cannot be written in the form of p/q, q is not equal to zero, and p and q are integers.
   
• Long division method: A method used to find the square root of non-perfect square numbers by systematically dividing.
   
• Approximation method: A method used to estimate the square root of a number by identifying two perfect squares between which the number lies and using a formula to find an approximate value.