124 LearnersLast updated on May 26th, 2025
Square Root of 221
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 221.
What is the Square Root of 221?
The square root is the inverse of the square of the number. 221 is not a perfect square. The square root of 221 is expressed in both radical and exponential form. In the radical form, it is expressed as √221, whereas (221)^(1/2) in the exponential form. √221 ≈ 14.86607, 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 221
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 method and approximation method are used. Let us now learn the following methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 221 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 221 is broken down into its prime factors:
Step 1: Finding the prime factors of 221 Breaking it down, we get 13 x 17.
Step 2: Now we found out the prime factors of 221. The second step is to make pairs of those prime factors. Since 221 is not a perfect square, the digits of the number can’t be grouped in pairs. Therefore, calculating 221 using prime factorization is impossible.
Square Root of 221 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 221, we need to group it as 21 and 2.
Step 2: Now we need to find n whose square is less than or equal to 2. We can say n is ‘1’ because 1 x 1 is less than or equal to 2. Now the quotient is 1, and after subtracting 1 from 2, the remainder is 1.
Step 3: Now let us bring down 21, 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, and we need to find the value of n.
Step 5: The next step is finding 2n × n ≤ 121. Let us consider n as 5, now 25 x 5 = 125, which is too large. Trying n as 4, we have 24 x 4 = 96, which is less than 121.
Step 6: Subtract 96 from 121, and the difference is 25. The quotient is now 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 2500.
Step 8: Now we need to find the new divisor, which is 298, because 298 x 8 = 2384.
Step 9: Subtracting 2384 from 2500 we get the result 116. 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 √221 is approximately 14.86.
Square Root of 221 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. Now let us learn how to find the square root of 221 using the approximation method.
Step 1: Now we have to find the closest perfect squares around √221. The smallest perfect square less than 221 is 196 (14^2) and the largest perfect square greater than 221 is 225 (15^2). √221 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) Using the formula (221 - 196) ÷ (225 - 196) = 25 ÷ 29 ≈ 0.86 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 + 0.86 = 14.86, so the square root of 221 is approximately 14.86.
Common Mistakes and How to Avoid Them in the Square Root of 221
Students do make mistakes while finding the square root, like forgetting about the negative square root, 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.
Mistake 2
Not adding 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 keep away from 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 221 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √221?
The area of the square is 221 square units.
Explanation
The area of the square = side^2.
The side length is given as √221.
Area of the square = side^2 = √221 x √221 = 221.
Therefore, the area of the square box is 221 square units.
Problem 2
A square-shaped building measuring 221 square feet is built; if each of the sides is √221, what will be the square feet of half of the building?
110.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 221 by 2, we get 110.5.
So half of the building measures 110.5 square feet.
Problem 3
Calculate √221 x 5.
74.33
Explanation
The first step is to find the square root of 221, which is approximately 14.86.
The second step is to multiply 14.86 with 5. So 14.86 x 5 = 74.33.
Problem 4
What will be the square root of (208 + 13)?
The square root is 15.
Explanation
To find the square root, we need to find the sum of (208 + 13).
208 + 13 = 221, and then √221 ≈ 14.86.
Therefore, the square root of (208 + 13) is approximately ±14.86.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √221 units and the width ‘w’ is 30 units.
Perimeter of the rectangle is approximately 89.72 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√221 + 30) = 2 × (14.86 + 30) = 2 × 44.86 = 89.72 units.
FAQ on Square Root of 221
1.What is √221 in its simplest form?
2.Mention the factors of 221.
3.Calculate the square of 221.
4.Is 221 a prime number?
5.221 is divisible by?
6.How does learning Algebra help students in Thailand make better decisions in daily life?
7.How can cultural or local activities in Thailand support learning Algebra topics such as Square Root of 221?
8.How do technology and digital tools in Thailand support learning Algebra and Square Root of 221?
9.Does learning Algebra support future career opportunities for students in Thailand?
Important Glossaries for the Square Root of 221
- Square root: A square root of a number is a value that, when multiplied by itself, gives the original number. Example: 4^2 = 16 and the inverse of the square is the square root, so √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 is expressing a number as the product of its prime numbers. Example: 221 = 13 x 17.
- 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.
Explore More algebra
About BrightChamps in Thailand
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.
