124 LearnersLast updated on May 26th, 2025
Square Root of 82
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 various fields such as engineering, physics, and mathematics. Here, we will discuss the square root of 82.
What is the Square Root of 82?
The square root is the inverse of the square of the number. 82 is not a perfect square. The square root of 82 is expressed in both radical and exponential form.
In the radical form, it is expressed as √82, whereas (82)(1/2) in the exponential form. √82 ≈ 9.05539, 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 82
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 82 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Let's look at how 82 is broken down into its prime factors:
Step 1: Finding the prime factors of 82 Breaking it down, we get 2 x 41, which are both prime numbers.
Step 2: Now that we have found the prime factors of 82, we see that the number cannot be grouped into pairs.
Therefore, calculating 82 using prime factorization alone as a perfect square is impossible.
Square Root of 82 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 82, we group it as 82.
Step 2: Now, we need to find n whose square is less than or equal to 82. We can say n as ‘9’ because 9 x 9 = 81.
Step 3: The quotient is 9, and after subtracting 81 from 82, the remainder is 1.
Step 4: Since the remainder is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeros to the remainder. Now the new dividend is 100.
Step 5: The new divisor is 18 (double of 9), and we need to find the next digit of the quotient. We choose 5 because 185 x 5 = 925.
Step 6: Subtract 925 from 1000 to get 75.
Step 7: Continue doing these steps until we get two decimal places.
So the square root of √82 ≈ 9.05
Square Root of 82 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 82 using the approximation method.
Step 1: Now we have to find the closest perfect square of √82. The smallest perfect square less than 82 is 81, and the largest perfect square greater than 82 is 100. √82 falls between 9 and 10.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).
Using the formula: (82 - 81) / (100 - 81) = 1 / 19 ≈ 0.0526 Adding this to the base square root, 9 + 0.0526 = 9.0526,
so the square root of 82 is approximately 9.05.
Common Mistakes and How to Avoid Them in the Square Root of 82
Students do make mistakes while finding the square root, like forgetting about the negative square root or skipping steps in methods. 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 has both positive and negative square roots. However, in most practical applications, we only consider the positive square root. For example, √82 ≈ 9.055, but there is also -9.055, which should not be overlooked.
Square Root of 82 Examples
Problem 1
Can you help Alex find the area of a square box if its side length is given as √82?
The area of the square is 82 square units.
Explanation
The area of the square = side².
The side length is given as √82.
Area of the square = side² = √82 x √82 = 82.
Therefore, the area of the square box is 82 square units.
Problem 2
A square-shaped plot measuring 82 square feet is built; if each of the sides is √82, what will be the square feet of half of the plot?
41 square feet
Explanation
We can just divide the given area by 2 as the plot is square-shaped.
Dividing 82 by 2, we get 41.
So half of the plot measures 41 square feet.
Problem 3
Calculate √82 x 4.
36.22
Explanation
The first step is to find the square root of 82, which is approximately 9.055.
The second step is to multiply 9.055 by 4. So 9.055 x 4 ≈ 36.22.
Problem 4
What will be the square root of (77 + 5)?
The square root is 9.
Explanation
To find the square root, we need to find the sum of (77 + 5). 77 + 5 = 82, and then √82 ≈ 9.055.
Therefore, the square root of (77 + 5) is approximately 9.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √82 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as approximately 98.11 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√82 + 40)
= 2 × (9.055 + 40) ≈ 2 × 49.055 ≈ 98.11 units.
FAQ on Square Root of 82
1.What is √82 in its simplest form?
2.Mention the factors of 82.
3.Calculate the square of 82.
4.Is 82 a prime number?
5.82 is divisible by?
6.How does learning Algebra help students in United Kingdom make better decisions in daily life?
7.How can cultural or local activities in United Kingdom support learning Algebra topics such as Square Root of 82?
8.How do technology and digital tools in United Kingdom support learning Algebra and Square Root of 82?
9.Does learning Algebra support future career opportunities for students in United Kingdom?
Important Glossaries for the Square Root of 82
- Square root: A square root is the inverse of squaring a number. Example: 3² = 9 and the inverse of the square is the square root that is √9 = 3.
- Irrational number: An irrational number is a number that cannot be expressed as a simple fraction; its decimal representation is non-repeating and non-terminating.
- Perfect square: A perfect square is a number that is the square of an integer. Example: 4, 9, and 16 are perfect squares.
- Radical: A radical is a symbol (√) that represents the root of a number.
- Approximation: An approximation is a value or number that is close to but not exactly equal to the exact value.
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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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