103 LearnersLast updated on May 26th, 2025
Square Root of 585
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 585.
What is the Square Root of 585?
The square root is the inverse of the square of the number. 585 is not a perfect square. The square root of 585 is expressed in both radical and exponential form. In the radical form, it is expressed as √585, whereas (585)^(1/2) in the exponential form. √585 ≈ 24.18677, 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 585
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 585 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 585 is broken down into its prime factors.
Step 1: Finding the prime factors of 585. Breaking it down, we get 3 x 3 x 5 x 13: 3^2 x 5^1 x 13^1
Step 2: Now that we found the prime factors of 585, the second step is to make pairs of those prime factors. Since 585 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 585 using prime factorization is not straightforward.
Square Root of 585 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 585, we need to group it as 85 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 x 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 85, 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 quotient and the new digit. Now we get 4n as the new divisor, we need to find the value of n.
Step 5: The next step is finding 4n × n ≤ 185. Let us consider n as 4, now 44 x 4 = 176.
Step 6: Subtract 176 from 185, the difference is 9, and the quotient is 24.
Step 7: Since the dividend is less 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 900.
Step 8: Now we need to find a new n such that the new divisor (481) multiplied by this n is less than or equal to 900. Suppose n is 1, then 481 x 1 = 481.
Step 9: Subtracting 481 from 900, we get the result 419.
Step 10: Now the quotient is 24.1
Step 11: 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 √585 ≈ 24.18.
Square Root of 585 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 585 using the approximation method.
Step 1: Now we have to find the closest perfect squares to √585. The smallest perfect square less than 585 is 576, and the largest perfect square greater than 585 is 625. √585 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 (585 - 576) / (625 - 576) = 0.18. 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 24 + 0.18 = 24.18, so the square root of 585 is approximately 24.18.
Common Mistakes and How to Avoid Them in the Square Root of 585
Students do make mistakes while finding the square root, like forgetting about the negative square root or 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.
Square Root of 585 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √585?
The area of the square is approximately 585 square units.
Explanation
The area of the square = side^2.
The side length is given as √585.
Area of the square = side^2 = √585 x √585 = 585.
Therefore, the area of the square box is approximately 585 square units.
Problem 2
A square-shaped building measuring 585 square feet is built; if each of the sides is √585, what will be the square feet of half of the building?
292.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 585 by 2 = we get 292.5.
So half of the building measures 292.5 square feet.
Problem 3
Calculate √585 x 5.
Approximately 120.93385
Explanation
The first step is to find the square root of 585, which is approximately 24.18677.
The second step is to multiply 24.18677 with 5.
So 24.18677 x 5 ≈ 120.93385.
Problem 4
What will be the square root of (585 + 15)?
The square root is approximately 24.837.
Explanation
To find the square root, we need to find the sum of (585 + 15). 585 + 15 = 600, and then the square root of 600 is approximately 24.495.
Therefore, the square root of (585 + 15) is approximately ±24.495.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √585 units and the width ‘w’ is 40 units.
We find the perimeter of the rectangle as approximately 128.37354 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√585 + 40) = 2 × (24.18677 + 40) = 2 × 64.18677 ≈ 128.37354 units.
FAQ on Square Root of 585
1.What is √585 in its simplest form?
2.Mention the factors of 585.
3.Calculate the square of 585.
4.Is 585 a prime number?
5.585 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 585?
8.How do technology and digital tools in India support learning Algebra and Square Root of 585?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 585
- Square root: A square root is the inverse of a square. Example: 4^2 = 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.
- Non-perfect square: A non-perfect square is a number that does not have an integer as its square root, like 585.
- Long division method: A step-by-step method to find the square root of numbers that are not perfect squares.
- Prime factorization: Expressing a number as a product of its prime factors, used to simplify square roots.
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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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