130 LearnersLast updated on May 26th, 2025
Square Root of 509
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, etc. Here, we will discuss the square root of 509.
What is the Square Root of 509?
The square root is the inverse of the square of a number. 509 is not a perfect square. The square root of 509 is expressed in both radical and exponential form. In the radical form, it is expressed as √509, whereas (509)^(1/2) in the exponential form. √509 ≈ 22.549, 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 509
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 and approximation methods are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 509 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 509 is broken down into its prime factors.
Step 1: Finding the prime factors of 509
Breaking it down, we get 7 x 73: 7^1 x 73^1
Step 2: Now that we have found the prime factors of 509, the next step is to make pairs of those prime factors. Since 509 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating √509 using prime factorization is not feasible.
Square Root of 509 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 509, we need to group it as 09 and 5.
Step 2: Now we need to find n whose square is 5. We can say n is ‘2’ because 2 x 2 = 4, which is lesser 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 09, which is the new dividend. Add the old divisor with the same number, 2 + 2, to get 4, which will be our new divisor.
Step 4: The new divisor will be the sum of the previous divisor 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 ≤ 109. Let us consider n as 2, now 4 x 2 x 2 = 88.
Step 6: Subtract 88 from 109, and the difference is 21.
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 2100.
Step 8: Now we need to find the new divisor, which is 45, because 445 x 5 = 2225.
Step 9: Subtracting 2225 from 2100, we get the result -125.
Step 10: Now the quotient is 22.5.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal value, continue until the remainder is zero.
So the square root of √509 ≈ 22.54
Square Root of 509 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 509 using the approximation method.
Step 1: Now we have to find the closest perfect square of √509. The smallest perfect square less than 509 is 484, and the largest perfect square greater than 509 is 529. √509 falls somewhere between 22 and 23.
Step 2: Now we need to apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (509 - 484) / (529 - 484) = 0.56
Using the formula, we identified the decimal part of our square root. The next step is adding the value we got initially to the decimal number, which is 22 + 0.56 = 22.56, so the square root of 509 is approximately 22.56.
Common Mistakes and How to Avoid Them in the Square Root of 509
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping long division steps. 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, we usually consider 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 symbol 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 before moving on to the next step.
For example: √(12 + 24) ≠ √12 + √24. The correct way 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 incorrect; the correct answer is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root consistently, 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 calculation. This may occur in the subtraction step or elsewhere.
Square Root of 509 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √509?
The area of the square is 509 square units.
Explanation
The area of the square = side^2.
The side length is given as √509.
Area of the square = side^2 = √509 x √509 = 509.
Therefore, the area of the square box is 509 square units.
Problem 2
A square-shaped building measuring 509 square feet is built; if each of the sides is √509, what will be the square feet of half of the building?
254.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 509 by 2 = we get 254.5.
So half of the building measures 254.5 square feet.
Problem 3
Calculate √509 x 5.
112.745
Explanation
The first step is to find the square root of 509, which is approximately 22.549, and the second step is to multiply 22.549 by 5.
So 22.549 x 5 ≈ 112.745.
Problem 4
What will be the square root of (500 + 9)?
The square root is approximately 22.54.
Explanation
To find the square root, we need to find the sum of (500 + 9). 500 + 9 = 509, and then √509 ≈ 22.54.
Therefore, the square root of (500 + 9) is approximately ±22.54.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √509 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 121.098 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√509 + 38) = 2 × (22.549 + 38) ≈ 2 × 60.549 ≈ 121.098 units.
FAQ on Square Root of 509
1.What is √509 in its simplest form?
2.Is 509 a perfect square?
3.Calculate the square of 509.
4.Is 509 a prime number?
5.509 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 509?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 509?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 509
- Square root: A square root is a value that, when multiplied by itself, gives the original number. For example, 5^2 = 25, and the inverse of the square is the square root, that is √25 = 5.
- 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.
- Prime number: A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. For example, 2, 3, 5, 7, etc.
- Long division method: A traditional method used to find the square root of a non-perfect square number by systematically dividing and averaging.
- Approximation method: A method used to estimate the square root of a number by identifying the closest perfect squares and calculating a fractional approximation.
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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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