124 LearnersLast updated on May 26th, 2025
Square Root of 209
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 209.
What is the Square Root of 209?
The square root is the inverse of the square of the number. 209 is not a perfect square. The square root of 209 is expressed in both radical and exponential form. In the radical form, it is expressed as √209, whereas (209)^(1/2) in the exponential form. √209 ≈ 14.4568, 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 209
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 209 by Prime Factorization Method
The product of prime factors is the Prime factorization of a number. Now let us look at how 209 is broken down into its prime factors.
Step 1: Finding the prime factors of 209 Breaking it down, we get 11 × 19.
Step 2: Now we found out the prime factors of 209. The second step is to make pairs of those prime factors. Since 209 is not a perfect square, the digits of the number can’t be grouped in pairs.
Therefore, calculating 209 using prime factorization is not practical for finding its square root.
Square Root of 209 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 209, we need to group it as 09 and 2.
Step 2: Now we need to find n whose square is less than or equal to 2. We can say n as ‘1’ because 1 × 1 is less than or equal to 2. Now the quotient is 1, and after subtracting 1² from 2, 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 1 + 1 to get 2, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we have 2n as the new divisor; we need to find the value of n.
Step 5: Finding 2n × n ≤ 109, let us consider n as 4. Now 24 × 4 = 96.
Step 6: Subtract 96 from 109; the difference is 13, and the quotient is 14.
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 1300.
Step 8: Now we need to find the new divisor that is 289 because 289 × 4 = 1156.
Step 9: Subtracting 1156 from 1300, we get the result 144. Step 10: Now the quotient is 14.4.
Step 11: Continue these steps until we get two numbers after the decimal point or until the remainder is zero.
So the square root of √209 is approximately 14.46.
Square Root of 209 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. Let us learn how to find the square root of 209 using the approximation method.
Step 1: We have to find the closest perfect squares of √209. The smallest perfect square less than 209 is 196, and the largest perfect square greater than 209 is 225. √209 falls somewhere between 14 and 15.
Step 2: Now apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (209 - 196) / (225 - 196) = 13 / 29 ≈ 0.448. Adding this to 14, we get 14 + 0.448 = 14.448, so the square root of 209 is approximately 14.45.
Common Mistakes and How to Avoid Them in the Square Root of 209
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in methods like long division. Let's look at a few mistakes students tend to make in detail.
Mistake 1
Forgetting about the negative square root
It is important to be aware that a number has both positive and negative square roots. However, we typically take only the positive square root, as it is the required one.
For example: √50 ≈ 7.07, but there is also -7.07 which should not be forgotten.
Mistake 2
Not adding square root symbol in proper places
Not using the square root symbol properly is a common mistake. Teach students to simplify numbers inside the square root before moving on.
For example: √(12 + 24) ≠ √12 + √24. The correct way is √(12 + 24) = √36 = 6.
Mistake 3
Not finding the correct value of a non-perfect square
Students should first simplify numbers inside the square root to avoid mistakes in identifying the approximate value.
For example: √50 ≠ 7; the correct approximation is √50 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students must understand the difference between cube root and square root.
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, and students may skip a step accidentally, resulting in a wrong answer. This can happen during subtraction or other steps.
Square Root of 209 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √209?
The area of the square is approximately 209 square units.
Explanation
The area of the square = side².
The side length is given as √209.
Area of the square = (√209)² = 209.
Therefore, the area of the square box is approximately 209 square units.
Problem 2
A square-shaped building measuring 209 square feet is built; if each of the sides is √209, what will be the square feet of half of the building?
Approximately 104.5 square feet.
Explanation
Divide the given area by 2, as the building is square-shaped.
Dividing 209 by 2 gives approximately 104.5.
So half of the building measures approximately 104.5 square feet.
Problem 3
Calculate √209 × 5.
Approximately 72.28.
Explanation
The first step is to find the square root of 209, which is approximately 14.46.
The second step is to multiply 14.46 with 5. So 14.46 × 5 ≈ 72.28.
Problem 4
What will be the square root of (200 + 9)?
The square root is 14.46.
Explanation
To find the square root, we need to find the sum of (200 + 9). 200 + 9 = 209, and then √209 ≈ 14.46. Therefore, the square root of (200 + 9) is approximately ±14.46.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √209 units and the width ‘w’ is 40 units.
The perimeter of the rectangle is approximately 108.92 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√209 + 40) ≈ 2 × (14.46 + 40) ≈ 2 × 54.46 ≈ 108.92 units.
FAQ on Square Root of 209
1.What is √209 in its simplest form?
2.Mention the factors of 209.
3.Calculate the square of 209.
4.Is 209 a prime number?
5.Is 209 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 209?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 209?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 209
- Square root: The square root is the inverse operation of squaring a number. For example, 4² = 16, and the inverse is the square root: √16 = 4.
- Irrational number: An irrational number cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
- Principal square root: A number has both positive and negative square roots, but the positive square root is typically used, known as the principal square root.
- Perfect square: A perfect square is a number that is the square of an integer. For example, 144 is a perfect square because it is 12².
- Approximation: The process of finding a value close to the actual value, used especially when dealing with irrational numbers or non-perfect squares.
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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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