121 LearnersLast updated on May 26th, 2025
Square Root of 682
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 682.
What is the Square Root of 682?
The square root is the inverse of the square of the number. 682 is not a perfect square. The square root of 682 is expressed in both radical and exponential form. In radical form, it is expressed as √682, whereas (682)^(1/2) in exponential form. √682 ≈ 26.0998, 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 682
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 682 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 682 is broken down into its prime factors.
Step 1: Finding the prime factors of 682 Breaking it down, we get 2 × 11 × 31.
Step 2: Now we found out the prime factors of 682. The second step is to make pairs of those prime factors. Since 682 is not a perfect square, therefore the digits of the number can’t be grouped in pairs.
Therefore, calculating 682 using prime factorization is not straightforward for finding the exact square root.
Square Root of 682 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 682, we need to group it as 82 and 6.
Step 2: Now we need to find n whose square is less than or equal to 6. We can say n is ‘2’ because 2 × 2 = 4 is less than 6. Now the quotient is 2, and after subtracting 4 from 6, the remainder is 2.
Step 3: Bring down 82, making the new dividend 282. Add the old divisor (2) to itself to get 4, which will be part of our new divisor.
Step 4: The new divisor will be 4n. We need to find the value of n such that 4n × n ≤ 282. Let n be 6, then 46 × 6 = 276.
Step 5: Subtract 276 from 282, getting a remainder of 6, and the quotient is 26.
Step 6: Since the dividend is less than the divisor, add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend, making it 600.
Step 7: The new divisor becomes 52. We find n such that 52n × n ≤ 600. Suppose n is 1, then 521 × 1 = 521.
Step 8: Subtract 521 from 600, the difference is 79.
Step 9: Continue this process to achieve the desired decimal places.
So the square root of √682 ≈ 26.0998.
Square Root of 682 by Approximation Method
The approximation method is another method for finding square roots, and 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 682 using the approximation method.
Step 1: Find the closest perfect squares of √682. The closest perfect square less than 682 is 676, and the closest perfect square greater than 682 is 729. √682 falls between 26 and 27.
Step 2: Apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula: (682 - 676) ÷ (729 - 676) = 6 ÷ 53 ≈ 0.1132 Add this decimal to the lower integer value, 26 + 0.1132 ≈ 26.1132.
So the square root of 682 is approximately 26.1132.
Common Mistakes and How to Avoid Them in the Square Root of 682
Students do make mistakes while finding the square root, such as forgetting about the negative square root and 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 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 ≈ 7.071067.
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 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 either happen in subtracting or elsewhere.
Square Root of 682 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √682?
The area of the square is approximately 465.6404 square units.
Explanation
The area of the square = side².
The side length is given as √682.
Area of the square = side² = √682 × √682 ≈ 26.0998 × 26.0998 ≈ 681.9996
Therefore, the area of the square box is approximately 682 square units.
Problem 2
A square-shaped building measuring 682 square feet is built; if each of the sides is √682, what will be the square feet of half of the building?
341 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 682 by 2 = we get 341.
So half of the building measures 341 square feet.
Problem 3
Calculate √682 × 5.
130.499
Explanation
The first step is to find the square root of 682, which is approximately 26.0998.
The second step is to multiply 26.0998 by 5.
So 26.0998 × 5 ≈ 130.499.
Problem 4
What will be the square root of (682 + 18)?
The square root is 26.
Explanation
To find the square root, we need to find the sum of (682 + 18).
682 + 18 = 700, and then √700 ≈ 26.4575.
Therefore, the square root of (682 + 18) is approximately 26.4575.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √682 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 130.1996 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√682 + 38)
≈ 2 × (26.0998 + 38)
≈ 2 × 64.0998
≈ 128.1996 units.
FAQ on Square Root of 682
1.What is √682 in its simplest form?
2.Mention the factors of 682.
3.Calculate the square of 682.
4.Is 682 a prime number?
5.682 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 682?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 682?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 682
- Square root: A square root is the inverse of a square. Example: 4² = 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.
- Prime factorization: Expressing a number as the product of its prime factors is prime factorization.
- Long division method: A method used to find the square root of non-perfect squares by dividing and averaging over several steps.
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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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