121 LearnersLast updated on May 26th, 2025
Square Root of 208
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 208.
What is the Square Root of 208?
The square root is the inverse of the square of the number. 208 is not a perfect square. The square root of 208 is expressed in both radical and exponential form. In the radical form, it is expressed as √208, whereas 208^(1/2) in the exponential form. √208 ≈ 14.422, 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 208
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 208 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 208 is broken down into its prime factors:
Step 1: Finding the prime factors of 208 Breaking it down, we get 2 x 2 x 2 x 2 x 13: 2^4 x 13
Step 2: Now we found out the prime factors of 208. The second step is to make pairs of those prime factors. Since 208 is not a perfect square, the digits of the number can’t be grouped in pairs completely.
Therefore, calculating 208 using prime factorization only gives an approximation.
Square Root of 208 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 208, we need to group it as 08 and 2.
Step 2: Now we need to find n whose square is 1. We can say n as ‘1’ because 1 x 1 is lesser than or equal to 2. Now the quotient is 1, and after subtracting 1 x 1 from 2, the remainder is 1.
Step 3: Now let us bring down 08, which is the new dividend. Add the old divisor with the same number: 1 + 1, which gives 2, 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 x n ≤ 108. Let us consider n as 4, now 24 x 4 = 96.
Step 6: Subtract 108 from 96; the difference is 12, 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 1200.
Step 8: Now we need to find the new divisor: 289, because 289 x 4 = 1156.
Step 9: Subtracting 1156 from 1200, we get the result 44.
Step 10: Now the quotient is 14.4
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose there are no decimal values; continue till the remainder is zero.
So the square root of √208 is approximately 14.42.
Square Root of 208 by Approximation Method
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 208 using the approximation method.
Step 1: Now we have to find the closest perfect square of √208. The smallest perfect square near 208 is 196, and the largest perfect square near 208 is 225. √208 falls somewhere between 14 and 15.
Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Largest perfect square - smallest perfect square). Going by the formula (208 - 196) / (225 - 196) = 0.41 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: 14 + 0.41 = 14.41, so the square root of 208 is approximately 14.41
Common Mistakes and How to Avoid Them in the Square Root of 208
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 a 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 ≈ 7.071.
Mistake 4
Confusing the square root symbol with the cube root
Students need to be taught the difference between cube root and square root 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 208 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √208?
The area of the square is approximately 208 square units.
Explanation
The area of the square = side².
The side length is given as √208.
Area of the square = side² = √208 x √208 = 208.
Therefore, the area of the square box is approximately 208 square units.
Problem 2
A square-shaped building measuring 208 square feet is built; if each of the sides is √208, what will be the square feet of half of the building?
104 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 208 by 2, we get 104.
So half of the building measures 104 square feet.
Problem 3
Calculate √208 x 5.
Approximately 72.11
Explanation
The first step is to find the square root of 208, which is approximately 14.42.
The second step is to multiply 14.42 with 5. So, 14.42 x 5 ≈ 72.11
Problem 4
What will be the square root of (196 + 12)?
The square root is approximately 14.28
Explanation
To find the square root, we need to find the sum of (196 + 12). 196 + 12 = 208, and then √208 ≈ 14.28.
Therefore, the square root of (196 + 12) is approximately ±14.28.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √208 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 104.84 units.
Explanation
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√208 + 38) ≈ 2 × (14.42 + 38) ≈ 2 × 52.42 ≈ 104.84 units.
FAQ on Square Root of 208
1.What is √208 in its simplest form?
2.Mention the factors of 208.
3.Calculate the square of 208.
4.Is 208 a prime number?
5.208 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 208?
8.How do technology and digital tools in Indonesia support learning Algebra and Square Root of 208?
9.Does learning Algebra support future career opportunities for students in Indonesia?
Important Glossaries for the Square Root of 208
- 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: It is the expression of a number as the product of its prime factors. Example: The prime factorization of 208 is 2^4 x 13.
- 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.
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