Square Root of 722

The square root is the inverse of the square of the number. 722 is not a perfect square. The square root of 722 is expressed in both radical and exponential form. In the radical form, it is expressed as √722, whereas (722)^(1/2) in the exponential form. √722 ≈ 26.85144, 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 722 is broken down into its prime factors.

Step 1: Finding the prime factors of 722

Breaking it down, we get 2 x 361, and 361 can be further broken down into 19 x 19: 2 x 19^2

Step 2: Now we found out the prime factors of 722. The second step is to make pairs of those prime factors. Since 722 is not a perfect square, therefore the digits of the number can’t be grouped in pairs perfectly. Therefore, calculating 722 using prime factorization gives an approximate result.

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 722, we need to group it as 22 and 7.

Step 2: Now we need to find n whose square is less than or equal to 7. We can say n is ‘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 22, making the new dividend 322. Add the old divisor with the same number 2 + 2 to get 4, which will be our new divisor.

Step 4: We now have 4n as the new divisor; we need to find n such that 4n x n is less than or equal to 322. By trying n = 6, we get 46 x 6 = 276.

Step 5: Subtract 276 from 322, getting a remainder of 46. The quotient is now 26.

Step 6: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to bring down two zeroes to the dividend. Now the new dividend is 4600.

Step 7: Now we need to find the new divisor that is 268 because 2688 x 8 = 4304.

Step 8: Subtracting 4304 from 4600, we get a remainder of 296.

Step 9: Now the quotient is 26.8

Step 10: 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 √722 is approximately 26.85.

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 722 using the approximation method.

Step 1: Now we have to find the closest perfect squares around √722. The smallest perfect square less than 722 is 676 (26^2) and the largest perfect square greater than 722 is 729 (27^2). √722 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 (722 - 676) ÷ (729 - 676) = 46 / 53 ≈ 0.8679. 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 + 0.8679 ≈ 26.8679, so the square root of 722 is approximately 26.85.

• Square root: A square root is the inverse of a square. Example: 4^2 = 16 and the inverse 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.

• Perfect square: A perfect square is an integer that is the square of an integer. For example, 36 is a perfect square because it is 6^2.

• 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.

• Approximation: Approximation is the process of finding a number close to the exact value, often used when dealing with irrational numbers.