102 LearnersLast updated on May 26th, 2025
Square Root of 586
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 like vehicle design, finance, etc. Here, we will discuss the square root of 586.
What is the Square Root of 586?
The square root is the inverse of the square of a number. 586 is not a perfect square. The square root of 586 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √586, whereas (586)^(1/2) is the exponential form. √586 ≈ 24.2074, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 586
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers, 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 586 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 586 is broken down into its prime factors.
Step 1: Finding the prime factors of 586 Breaking it down, we get 2 × 293: 2^1 × 293^1
Step 2: Now we found out the prime factors of 586. The second step is to make pairs of those prime factors. Since 586 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 586 using prime factorization for its square root is not straightforward.
Square Root of 586 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 to 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 586, we need to group it as 86 and 5.
Step 2: Now we need to find n whose square is less than or equal to 5. We can say n is 2 because 2 × 2 = 4 is less than or equal to 5. Now the quotient is 2, and after subtracting 4 from 5, the remainder is 1.
Step 3: Now let us bring down 86, which is the new dividend. Add the old divisor with the same number: 2 + 2 = 4, which will be our new divisor.
Step 4: We get the new divisor as 4n; we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 186. Let us consider n as 2; now 42 × 2 = 84.
Step 6: Subtract 186 from 84; the difference is 102, and the quotient is 22.
Step 7: Since the dividend is greater 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 10200.
Step 8: Now we need to find the new divisor, assuming n is 2 because 442 × 2 = 884.
Step 9: Subtracting 884 from 10200, we get the result 9316.
Step 10: Now the quotient is 24.2.
Step 11: Continue doing these steps until we achieve a desired level of precision, typically two decimal places.
So the square root of √586 ≈ 24.21.
Square Root of 586 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 586 using the approximation method.
Step 1: Now we have to find the closest perfect squares to √586. The smallest perfect square less than 586 is 576, and the largest perfect square greater than 586 is 625. Thus, √586 falls somewhere between 24 and 25.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (586 - 576) ÷ (625 - 576) = 10 ÷ 49 ≈ 0.2041. Adding this decimal to the smaller integer gives us the approximate square root: 24 + 0.2041 ≈ 24.2041. Thus, the square root of 586 is approximately 24.2041.
Common Mistakes and How to Avoid Them in the Square Root of 586
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division steps. Let's look at a few of these common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important to remember that a number has both positive and negative square roots. However, we usually take only the positive square root, as it is more commonly used in practical applications.
For example, for √50 ≈ 7.07, there is also -7.07, which should not be forgotten.
Square Root of 586 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √586?
The area of the square is approximately 586 square units.
Explanation
The area of the square = side^2.
The side length is given as √586.
Area of the square = side^2 = √586 × √586 = 586.
Therefore, the area of the square box is approximately 586 square units.
Problem 2
A square-shaped building measuring 586 square feet is built; if each of the sides is √586, what will be the square feet of half of the building?
293 square feet
Explanation
To find half of the building's area, simply divide the given area by 2. Dividing 586 by 2 = 293.
So half of the building measures 293 square feet.
Problem 3
Calculate √586 × 5.
Approximately 121.04
Explanation
First, find the square root of 586, which is approximately 24.21.
Then multiply 24.21 by 5. So 24.21 × 5 ≈ 121.04.
Problem 4
What will be the square root of (576 + 10)?
The square root is approximately 24.2
Explanation
To find the square root, calculate the sum of (576 + 10). 576 + 10 = 586, and then the square root of 586 is approximately 24.2.
Therefore, the square root of (576 + 10) is approximately ±24.2.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √586 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 124.42 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√586 + 38) = 2 × (24.21 + 38) = 2 × 62.21 = 124.42 units.
FAQ on Square Root of 586
1.What is √586 in its simplest form?
2.Mention the factors of 586.
3.Calculate the square of 586.
4.Is 586 a prime number?
5.586 is divisible by?
6.How does learning Algebra help students in Bahrain make better decisions in daily life?
7.How can cultural or local activities in Bahrain support learning Algebra topics such as Square Root of 586?
8.How do technology and digital tools in Bahrain support learning Algebra and Square Root of 586?
9.Does learning Algebra support future career opportunities for students in Bahrain?
Important Glossaries for the Square Root of 586
- Square root: A square root is a value that, when multiplied by itself, gives the original number. Example: 5^2 = 25, and the square root of 25 is √25 = 5.
- Irrational number: An irrational number cannot be expressed as a fraction of two integers. Its decimal form is non-repeating and non-terminating.
- Perfect square: A perfect square is a number that is the square of an integer. Example: 16 is a perfect square because 4^2 = 16.
- Long division method: A method used to find the square root of numbers, especially non-perfect squares, by dividing and averaging.
- Approximation: A technique to find a close estimate of a mathematical value, often used for non-perfect squares to determine their square roots.
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About BrightChamps in Bahrain
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.
