Square Root of 72.4

The square root is the inverse of the square of the number. 72.4 is not a perfect square. The square root of 72.4 is expressed in both radical and exponential form.

In the radical form, it is expressed as √72.4, whereas (72.4)^(1/2) in the exponential form. √72.4 ≈ 8.5047, 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 long-division method and approximation method are used. Let us now learn the following methods:

1. Prime factorization method
2. Long division method
3. Approximation method

The product of prime factors is the prime factorization of a number. Now let us look at how 72.4 is broken down into its prime factors.

Step 1: Finding the prime factors of 72.4 Since 72.4 is a decimal, we first multiply it by 10 to simplify. This gives us 724. Breaking it down, we get 2 x 2 x 181: 2² x 181¹

Step 2: Now we found out the prime factors of 724. The second step is to make pairs of those prime factors. Since 72.4 is not a perfect square, the digits of the number can’t be grouped in pairs.

Therefore, calculating 72.4 using prime factorization is not straightforward.

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 72.4, we consider it as 7240 for simplification.

Step 2: Now we need to find n whose square is closest to 72. We can say n is ‘8’ because 8 x 8 = 64, which is less than 72. Now the quotient is 8, after subtracting 64 from 72, the remainder is 8.

Step 3: Bring down 40, making the new dividend 840. Add the old divisor with the same number, 8 + 8 = 16, which will be our new divisor.

Step 4: The new divisor will be 16n. Find n such that 16n x n ≤ 840. Let us consider n as 5, now 165 x 5 = 825.

Step 5: Subtract 825 from 840, the difference is 15, and the quotient is 8.5.

Step 6: Since the dividend is less than the divisor, we add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 1500.

Step 7: Find the new divisor, which is 171, because 171 x 8 = 1368.

Step 8: Subtracting 1368 from 1500, the result is 132.

Step 9: 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 √72.4 is approximately 8.5047.

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 72.4 using the approximation method.

Step 1: Identify the closest perfect squares around 72.4. The closest perfect squares are 64 and 81, as √64 = 8 and √81 = 9.

Step 2: Now apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (72.4 - 64) / (81 - 64) = 0.4941.

Step 3: Add the decimal value to the smaller perfect square root. 8 + 0.5047 = 8.5047.

Hence, the square root of 72.4 is approximately 8.5047.

• 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.

• Decimal: A number that has a whole number and a fraction in a single number is called a decimal, for example, 7.86, 8.65, and 9.42 are decimals.

• Long Division Method: A process to find the square root of a number by dividing it into pairs of digits and finding the largest number whose square is less than or equal to the leftmost group.

• Approximation Method: A method to estimate the square root of a number by identifying two close perfect squares and estimating the root using their values.