135 LearnersLast updated on May 26th, 2025
Square Root of 72.4
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 72.4
What is the 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.
Finding the Square Root of 72.4
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:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 72.4 by Prime Factorization 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.
Square Root of 72.4 by Long Division Method
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.
Square Root of 72.4 by Approximation Method
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.
Common Mistakes and How to Avoid Them in the Square Root of 72.4
Students do make mistakes while finding the square root, like forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to make students aware that a number does have both positive and negative square roots. However, we will be taking only the positive square root, as it is the required one.
For example: √72.4 ≈ 8.5047, there is also -8.5047 which should not be forgotten.
Square root of 72.4 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √72.4?
The area of the square is approximately 524.47 square units.
Explanation
The area of the square = side2.
The side length is given as √72.4.
Area of the square = side2 = √72.4 x √72.4 ≈ 8.5047 x 8.5047 ≈ 72.4.
Therefore, the area of the square box is approximately 524.47 square units.
Problem 2
A square-shaped field measuring 72.4 square meters is built; if each of the sides is √72.4, what will be the square meters of half of the field?
36.2 square meters.
Explanation
We can just divide the given area by 2 as the field is square-shaped.
Dividing 72.4 by 2 = 36.2.
So half of the field measures 36.2 square meters.
Problem 3
Calculate √72.4 x 5.
Approximately 42.5235.
Explanation
The first step is to find the square root of 72.4, which is approximately 8.5047.
The second step is to multiply 8.5047 by 5.
So 8.5047 x 5 ≈ 42.5235.
Problem 4
What will be the square root of (64 + 8.4)?
The square root is approximately 8.6.
Explanation
To find the square root, we need to find the sum of (64 + 8.4). 64 + 8.4 = 72.4, and then √72.4 ≈ 8.5047.
Therefore, the square root of (64 + 8.4) is approximately ±8.6.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √72.4 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 93.01 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√72.4 + 38) ≈ 2 × (8.5047 + 38) ≈ 2 × 46.5047 ≈ 93.01 units.
FAQ on Square Root of 72.4
1.What is √72.4 in its simplest form?
2.Mention the factors of 72.4.
3.Calculate the square of 72.4.
4.Is 72.4 a prime number?
5.72.4 is divisible by?
6.How does learning Algebra help students in Oman make better decisions in daily life?
7.How can cultural or local activities in Oman support learning Algebra topics such as Square Root of 72.4?
8.How do technology and digital tools in Oman support learning Algebra and Square Root of 72.4?
9.Does learning Algebra support future career opportunities for students in Oman?
Important Glossaries for the Square Root of 72.4
- 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.
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
