Square Root of 680

The square root is the inverse of the square of the number. 680 is not a perfect square. The square root of 680 is expressed in both radical and exponential form. In the radical form, it is expressed as √680, whereas (680)^(1/2) in the exponential form. √680 = 26.0768, 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 680 is broken down into its prime factors.

Step 1: Finding the prime factors of 680 Breaking it down, we get 2 x 2 x 2 x 5 x 17: 2^3 x 5^1 x 17^1

Step 2: Now we found out the prime factors of 680. The second step is to make pairs of those prime factors. Since 680 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.

Therefore, calculating 680 using prime factorization 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 680, we need to group it as 80 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 after subtracting 6-4 the remainder is 2.

Step 3: Now let us bring down 80, 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 ≤ 280. Let us consider n as 6, now 4 x 6 x 6 = 276.

Step 6: Subtract 280 from 276, the difference is 4, 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 400.

Step 8: Now we need to find the new divisor that is 521, because 521 x 1 = 521.

Step 9: Subtracting 521 from 40000, we get the result 739.

Step 10: Now the quotient is 26.0.

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

So the square root of √680 is 26.08.

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 680 using the approximation method.

Step 1: Now we have to find the closest perfect square of √680. The smallest perfect square less than 680 is 676 and the largest perfect square more than 680 is 729. √680 falls somewhere between 26 and 27.

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 (680 - 676) ÷ (729-676) = 0.075 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 + 0.08 = 26.08, so the square root of 680 is 26.08.

• Square root: A square root is the inverse of squaring a number. For example: if 5^2 = 25, then the square root is √25 = 5.
   
• 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.
   
• Radical: A radical symbol (√) is used to denote the square root or nth root of a number.
   
• Decimal: A decimal is a way of representing numbers that include a whole number and a fractional part separated by a decimal point. For example: 3.14, 6.283.
   
• Long Division Method: A method used to divide larger numbers to find the quotient and remainder. It is often used to find square roots of non-perfect squares.