120 LearnersLast updated on May 26th, 2025
Square Root of 806
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 such as vehicle design, finance, and more. Here, we will discuss the square root of 806.
What is the Square Root of 806?
The square root is the inverse of the square of the number. 806 is not a perfect square. The square root of 806 is expressed in both radical and exponential form. In the radical form, it is expressed as √806, whereas (806)^(1/2) is the exponential form. √806 ≈ 28.403, 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 806
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 806 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 806 is broken down into its prime factors.
Step 1: Finding the prime factors of 806 Breaking it down, we get 2 x 13 x 31: 2^1 x 13^1 x 31^1
Step 2: Now we found out the prime factors of 806. The second step is to make pairs of those prime factors. Since 806 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 806 using prime factorization is impractical.
Square Root of 806 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 806, we need to group it as 06 and 8.
Step 2: Now we need to find n whose square is closest to 8. We can say n as ‘2’ because 2 x 2 = 4 is lesser than or equal to 8. Now the quotient is 2, after subtracting 8 - 4, the remainder is 4.
Step 3: Now let us bring down 06, which is the new dividend. Add the old divisor with the same number 2 + 2; we get 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, and we need to find the value of n.
Step 5: The next step is finding 4n x n ≤ 406. Let us consider n as 7, now 47 x 7 = 329.
Step 6: Subtract 406 from 329; the difference is 77, and the quotient is 27.
Step 7: 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 7700.
Step 8: Now we need to find the new divisor, which is 549 because 549 x 9 = 4941.
Step 9: Subtracting 4941 from 7700, we get the result 2759.
Step 10: Now the quotient is 28.4
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue till the remainder is zero.
So the square root of √806 is approximately 28.4.
Square Root of 806 by Approximation Method
The approximation method is another method for finding the 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 806 using the approximation method.
Step 1: Now we have to find the closest perfect square of √806.
The smallest perfect square less than 806 is 784, and the largest perfect square greater than 806 is 841. √806 falls somewhere between 28 and 29.
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 (806 - 784) ÷ (841 - 784) = 0.386
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 28 + 0.386 ≈ 28.39, so the square root of 806 is approximately 28.39.
Common Mistakes and How to Avoid Them in the Square Root of 806
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 = 7.07, and 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 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 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 806 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √706?
The area of the square is 705.636 square units.
Explanation
The area of the square = side^2.
The side length is given as √706.
Area of the square = side^2 = √706 x √706 ≈ 26.57 x 26.57 ≈ 705.636
Therefore, the area of the square box is approximately 705.636 square units.
Problem 2
A square-shaped building measuring 806 square feet is built; if each of the sides is √806, what will be the square feet of half of the building?
403 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 806 by 2 = we get 403.
So half of the building measures 403 square feet.
Problem 3
Calculate √806 x 5.
Approximately 142.015
Explanation
The first step is to find the square root of 806, which is approximately 28.403, and the second step is to multiply 28.403 by 5.
So 28.403 x 5 ≈ 142.015
Problem 4
What will be the square root of (706 + 6)?
The square root is 27.
Explanation
To find the square root, we need to find the sum of (706 + 6). 706 + 6 = 712, and then 712 ≈ 27.
Therefore, the square root of (706 + 6) is approximately ±27.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √706 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 129.14 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√706 + 38) ≈ 2 × (26.57 + 38) = 2 × 64.57 ≈ 129.14 units.
FAQ on Square Root of 806
1.What is √806 in its simplest form?
2.Mention the factors of 806.
3.Calculate the square of 806.
4.Is 806 a prime number?
5.806 is divisible by?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 806?
8.How do technology and digital tools in India support learning Algebra and Square Root of 806?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 806
- Square Root: A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since 4 x 4 = 16.
- Irrational Number: An irrational number cannot be expressed as a fraction p/q where p and q are integers and q ≠ 0. Examples include √2 and √3.
- Perfect Square: A perfect square is a number that is the square of an integer. For example, 36 is a perfect square because it is 6 x 6.
- Long Division Method: A method used to find the square root of non-perfect squares by dividing the number into groups and repeatedly finding remainders.
- Prime Factorization: The process of expressing a number as a product of its prime factors. This method is useful for simplifying roots and understanding number properties.
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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.
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