125 LearnersLast updated on May 26th, 2025
Square Root of 219
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 like vehicle design, finance, etc. Here, we will discuss the square root of 219.
What is the Square Root of 219?
The square root is the inverse of the square of the number. 219 is not a perfect square. The square root of 219 is expressed in both radical and exponential form. In the radical form, it is expressed as √219, whereas 219^(1/2) in the exponential form. √219 ≈ 14.798648, 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 219
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 219 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 219 is broken down into its prime factors:
Step 1: Finding the prime factors of 219 Breaking it down, we get 3 x 73: which is 3^1 x 73^1.
Step 2: Now we found out the prime factors of 219. The second step is to make pairs of those prime factors. Since 219 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 219 using prime factorization is impossible.
Square Root of 219 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 219, we need to group it as 19 and 2.
Step 2: Now we need to find n whose square is ≤ 2. We can say n is ‘1’ because 1 x 1 is less than or equal to 2. Now the quotient is 1; after subtracting 1 from 2, the remainder is 1.
Step 3: Now let us bring down 19, 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 get 2n as the new divisor; we need to find the value of n.
Step 5: The next step is finding 2n × n ≤ 119. Let us consider n as 4, now 24 x 4 = 96. Step 6: Subtract 96 from 119; the difference is 23, 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 zeros to the dividend. Now, the new dividend is 2300.
Step 8: Now we need to find the new divisor that is 148 because 1488 x 8 = 11904.
Step 9: Subtracting 11904 from 23000, we get the result 11096.
Step 10: Now the quotient is 14.8.
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 √219 is approximately 14.80.
Square Root of 219 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 219 using the approximation method.
Step 1: Now we have to find the closest perfect square of √219. The smallest perfect square less than 219 is 196, and the largest perfect square greater than 219 is 225. √219 falls somewhere between 14 and 15.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (219 - 196) ÷ (225 - 196) = 0.793. 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 14 + 0.793 = 14.793, so the square root of 219 is approximately 14.793.
Common Mistakes and How to Avoid Them in the Square Root of 219
Students make mistakes while finding the square root, such as forgetting about the negative square root or skipping the long division methods. Now let us look at a few of these 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 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 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 and then move on to the second step.
For example: √(12+24) = √12 + √24 is the wrong method of using roots. The right way of using roots 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 wrong; the correct answer is √50 ≈ 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root constantly, 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 sum. This may either happen in subtracting or elsewhere.
Square Root of 219 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √219?
The area of the square is approximately 219 square units.
Explanation
The area of the square = side^2.
The side length is given as √219.
Area of the square = side^2 = √219 x √219 = 219.
Therefore, the area of the square box is approximately 219 square units.
Problem 2
A square-shaped building measuring 219 square feet is built; if each of the sides is √219, what will be the square feet of half of the building?
109.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 219 by 2 = 109.5.
So half of the building measures 109.5 square feet.
Problem 3
Calculate √219 x 5.
Approximately 73.99
Explanation
The first step is to find the square root of 219, which is approximately 14.80.
The second step is to multiply 14.80 with 5. So 14.80 x 5 ≈ 74.00.
Problem 4
What will be the square root of (200 + 19)?
The square root is approximately 14.80
Explanation
To find the square root, we need to find the sum of (200 + 19). 200 + 19 = 219, and then √219 ≈ 14.80. Therefore, the square root of (200 + 19) is approximately ±14.80.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √219 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 105.60 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√219 + 38) = 2 × (14.80 + 38) ≈ 2 × 52.80 ≈ 105.60 units.
FAQ on Square Root of 219
1.What is √219 in its simplest form?
2.Mention the factors of 219.
3.Calculate the square of 219.
4.Is 219 a prime number?
5.219 is divisible by?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of 219?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 219?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 219
- 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.
- Principal square root: A number has both positive and negative square roots; however, it is always the positive square root that has more prominence due to its uses in the real world. That is the reason it is also known as the principal square root.
- Prime factorization: Prime factorization involves expressing a number as the product of its prime factors. For example, the prime factorization of 219 is 3 x 73.
- Decimal: If a number has a whole number and a fraction in a single number, then it is called a decimal. For example: 7.86, 8.65, and 9.42 are decimals.
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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.
Fun Fact
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