Square Root of 724

The square root is the inverse of the square of the number. 724 is not a perfect square. The square root of 724 is expressed in both radical and exponential forms. In the radical form, it is expressed as √724, whereas (724)^(1/2) in the exponential form. √724 ≈ 26.907, 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, it is not used for non-perfect square numbers, where the long-division method and approximation method are more suitable. 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 724 is broken down into its prime factors:

Step 1: Finding the prime factors of 724

Breaking it down, we get 2 × 2 × 181: 2^2 × 181^1

Step 2: Now we found the prime factors of 724. The second step is to make pairs of those prime factors. Since 724 is not a perfect square, the digits of the number can’t be grouped into pairs. Therefore, calculating 724 using prime factorization is not straightforward.

The long division method is particularly used for non-perfect square numbers. In this method, we 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 724, we need to group it as 24 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n as ‘2’ because 2 × 2 is 4, which is lesser than or equal to 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 24, 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; we need to find the value of n.

Step 5: The next step is finding 4n × n ≤ 324. Let us consider n as 6, now 46 × 6 = 276.

Step 6: Subtract 276 from 324; the difference is 48, 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 4800.

Step 8: Now we need to find the new divisor that is 9, because 539 × 9 = 4851.

Step 9: Subtracting 4851 from 4800, we get the result -51.

Step 10: Now the quotient is 26.9.

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 √724 is approximately 26.91.

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 724 using the approximation method.

Step 1: Now we have to find the closest perfect square of √724. The smallest perfect square less than 724 is 676, and the largest perfect square greater than 724 is 729. √724 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). Using the formula (724 - 676) ÷ (729 - 676) = 48 ÷ 53 ≈ 0.91. 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 + 0.91 = 26.91.

So the square root of 724 is approximately 26.91.

• 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.

• Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal; for example, 7.86, 8.65, and 9.42 are decimals.

• Long division method: A method used to find the square root of non-perfect square numbers by dividing the number into pairs starting from the decimal point and working towards the left.