121 LearnersLast updated on May 26th, 2025
Square Root of 722
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 722.
What is the 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.
Finding the Square Root of 722
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
Square Root of 722 by Prime Factorization 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.
Square Root of 722 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 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.
Square Root of 722 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 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.
Common Mistakes and How to Avoid Them in the Square Root of 722
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: √50 ≈ 7.071, there is also -7.071 which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root 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 a 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 ≈ 7.071067.
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 keep away from 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 problem. This may either happen in subtracting or elsewhere.
Square root of 722 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √722?
The area of the square is approximately 521.78 square units.
Explanation
The area of the square = side^2.
The side length is given as √722.
Area of the square = side^2 = √722 x √722 = 26.85 × 26.85 ≈ 721.6225.
Therefore, the area of the square box is approximately 721.62 square units.
Problem 2
A square-shaped building measuring 722 square feet is built; if each of the sides is √722, what will be the square feet of half of the building?
361 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 722 by 2, we get 361.
So half of the building measures 361 square feet.
Problem 3
Calculate √722 x 5.
134.25
Explanation
The first step is to find the square root of 722, which is approximately 26.85.
The second step is to multiply 26.85 by 5.
So 26.85 x 5 ≈ 134.25.
Problem 4
What will be the square root of (682 + 40)?
The square root is 27.
Explanation
To find the square root, we need to find the sum of (682 + 40) = 722, and then √722 ≈ 26.85.
Therefore, the square root of (682 + 40) is approximately ±26.85.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √722 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 129.7 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√722 + 38) ≈ 2 × (26.85 + 38) ≈ 2 × 64.85 ≈ 129.7 units.
FAQ on Square Root of 722
1.What is √722 in its simplest form?
2.Mention the factors of 722.
3.Calculate the square of 722.
4.Is 722 a prime number?
5.722 is divisible by?
6.How does learning Algebra help students in Indonesia make better decisions in daily life?
7.How can cultural or local activities in Indonesia support learning Algebra topics such as Square Root of 722?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 722?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 722
- 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.
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Jaskaran Singh Saluja
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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.
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