Square Root of 723

The square root is the inverse of the square of a number. 723 is not a perfect square. The square root of 723 is expressed in both radical and exponential form. In the radical form, it is expressed as √723, whereas (723)^(1/2) in the exponential form. √723 ≈ 26.851, 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 723 is broken down into its prime factors:

Step 1: Finding the prime factors of 723

Breaking it down, we get 3 x 241: 3^1 x 241^

Step 2: Now that we have found the prime factors of 723, the second step is to make pairs of those prime factors. Since 723 is not a perfect square, the digits of the number can’t be grouped in a pair. Therefore, calculating 723 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 723, we need to group it as 23 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 x 2 = 4 is less than 7. Now the quotient is 2, and after subtracting 4 from 7, the remainder is 3.

Step 3: Now let us bring down 23, 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 ≤ 323. Let us consider n as 8, now 48 x 8 = 384, which is more than 323, so we take n as 7, making 47 x 7 = 329.

Step 6: Subtract 329 from 323; we realize n = 6, making 46 x 6 = 276.

Step 7: Subtracting 276 from 323, the difference is 47, and the quotient is 26.8.

Step 8: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the dividend. Now the new dividend is 4700.

Step 9: The new divisor is 536, because 536 x 8 = 4288.

Step 10: Subtracting 4288 from 4700, we get the result 412.

Step 11: Now the quotient is 26.8.

Step 12: Continue doing these steps until we get two numbers after the decimal point, or continue until the remainder is zero.

So the square root of √723 is approximately 26.851.

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 723 using the approximation method.

Step 1: Find the closest perfect squares of √723. The smallest perfect square less than 723 is 676, and the largest perfect square greater than 723 is 729. √723 falls somewhere between 26 and 27.

Step 2: Now apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) Using the formula: (723 - 676) / (729 - 676) = 0.887 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 + 0.887 = 26.887, so the square root of 723 is approximately 26.887.

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

• Long division method: A method used to find the square root of non-perfect squares by grouping digits and using division.