126 LearnersLast updated on May 26th, 2025
Square Root of 724
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 fields like vehicle design, finance, etc. Here, we will discuss the square root of 724.
What is the 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.
Finding the Square Root of 724
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
Square Root of 724 by Prime Factorization 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.
Square Root of 724 by Long Division Method
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.
Square Root of 724 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 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.
Common Mistakes and How to Avoid Them in the Square Root of 724
Students do make mistakes while finding the square root, such as 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, √50 = 7.07; there is also -7.07, which should not be forgotten.
Mistake 2
Not adding the square root symbol in proper places
Not using the square root symbol properly is another mistake that children make. To resolve this, we need to teach children that simplifying the numbers inside the square root is important and then move on to the second step.
For example, √(12 + 24) = √12 + √24 is the wrong method of using roots.
The right way of using roots is √(12 + 24) = √36.
Mistake 3
Not finding the correct value of a non-perfect square
The first step that needs to be taught to children is to simplify the numbers inside the square root. This is important so that students do not make mistakes while identifying the approximate value.
For example, √50 = 7 is wrong; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly, as this will help them avoid mistakes.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in long division
Finding a square root using the long division method involves many steps. Due to this, many students might skip a step accidentally, which results in a wrong answer at the end of the sum. This may happen either in subtracting or elsewhere.
Square Root of 724 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √724?
The area of the square is approximately 524.76 square units.
Explanation
The area of the square = side^2.
The side length is given as √724.
Area of the square = side^2 = √724 × √724 ≈ 26.91 × 26.91 ≈ 724.
Therefore, the area of the square box is approximately 724 square units.
Problem 2
A square-shaped building measuring 724 square feet is built; if each of the sides is √724, what will be the square feet of half of the building?
362 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped. Dividing 724 by 2, we get 362. So half of the building measures 362 square feet.
Problem 3
Calculate √724 × 5.
Approximately 134.55
Explanation
The first step is to find the square root of 724, which is approximately 26.91.
The second step is to multiply 26.91 with 5. So 26.91 × 5 ≈ 134.55.
Problem 4
What will be the square root of (700 + 24)?
Approximately 26.91
Explanation
To find the square root, we need to find the sum of (700 + 24). 700 + 24 = 724, and then √724 ≈ 26.91.
Therefore, the square root of (700 + 24) is approximately ±26.91.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √724 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 129.82 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√724 + 38) ≈ 2 × (26.91 + 38) ≈ 2 × 64.91 ≈ 129.82 units.
FAQ on Square Root of 724
1.What is √724 in its simplest form?
2.Mention the factors of 724.
3.Calculate the square of 724.
4.Is 724 a prime number?
5.724 is divisible by?
Important Glossaries for the Square Root of 724
- 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.
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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.
