Square Root of 688

The square root is the inverse of the square of the number. 688 is not a perfect square. The square root of 688 is expressed in both radical and exponential form. In the radical form, it is expressed as √688, whereas (688)^(1/2) in the exponential form. √688 ≈ 26.229, which is an irrational number because it cannot be expressed in the form 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 methods 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 688 is broken down into its prime factors.

Step 1: Finding the prime factors of 688

Breaking it down, we get 2 x 2 x 2 x 43: 2^3 x 43^1

Step 2: Now we found out the prime factors of 688. The second step is to make pairs of those prime factors. Since 688 is not a perfect square, the digits of the number can’t be grouped in a pair. Therefore, calculating 688 using prime factorization to find a whole number is impossible.

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 688, we need to group it as 88 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 4 from 6, the remainder is 2.

Step 3: Now let us bring down 88, which is the new dividend. Add the old divisor with the same number 2 + 2 to 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, and we need to find the value of n.

Step 5: The next step is finding 4n x n ≤ 288. Let us consider n as 6; now 46 x 6 = 276.

Step 6: Subtracting 276 from 288, the difference is 12, and the quotient is 26.

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 1200.

Step 8: Now we need to find the new divisor, which is 524 because 524 x 2 = 1048.

Step 9: Subtracting 1048 from 1200, we get the result 152.

Step 10: Now the quotient is 26.2.

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 √688 is approximately 26.23.

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 688 using the approximation method.

Step 1: Now we have to find the closest perfect square of √688. The smallest perfect square less than 688 is 676, and the largest perfect square more than 688 is 729. √688 falls somewhere between 26 and 27.

Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square). Going by the formula (688 - 676) ÷ (729 - 676) = 12 ÷ 53 ≈ 0.23. 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 26 + 0.23 = 26.23, so the square root of 688 is approximately 26.23.

• 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: This is the process of expressing a number as a product of its prime factors.

• Long division method: A method used to find the square root of a non-perfect square by dividing the number into groups and solving step by step.