121 LearnersLast updated on May 26th, 2025
Square Root of 238
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 238
What is the Square Root of 238?
The square root is the inverse of the square of the number. 238 is not a perfect square. The square root of 238 is expressed in both radical and exponential form. In the radical form, it is expressed as √238, whereas (238)^(1/2) in the exponential form. √238 ≈ 15.4272, which is an irrational number because it cannot be expressed in the form p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 238
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 238 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 238 is broken down into its prime factors.
Step 1: Finding the prime factors of 238 Breaking it down, we get 2 x 7 x 17: 2^1 x 7^1 x 17^1
Step 2: Now we found out the prime factors of 238. The second step is to make pairs of those prime factors. Since 238 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating √238 using prime factorization is impossible.
Square Root of 238 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 238, we need to group it as 38 and 2.
Step 2: Now we need to find n whose square is 2. 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, the remainder is 1.
Step 3: Now let us bring down 38, which is the new dividend. Add the old divisor with the same number: 1 + 1, and we 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 x n ≤ 138. Let us consider n as 5. Now 25 x 5 = 125.
Step 6: Subtract 125 from 138, and the difference is 13, and the quotient is 15.
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 digit that makes 305n x n ≤ 1300. The digit is 4 because 3054 x 4 = 12216.
Step 9: Subtracting 12216 from 13000, we get the result 784.
Step 10: Now the quotient is 15.4.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there is no decimal values, continue till the remainder is zero. So the square root of √238 is approximately 15.42.
Square Root of 238 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 238 using the approximation method.
Step 1: Now we have to find the closest perfect squares of √238. The smallest perfect square less than 238 is 225, and the largest perfect square greater than 238 is 256. √238 falls somewhere between 15 and 16.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula: (238 - 225) / (256 - 225) = 13 / 31 ≈ 0.419. 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 15 + 0.419 = 15.419, so the square root of 238 is approximately 15.42.
Common Mistakes and How to Avoid Them in the Square Root of 238
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 the square root symbol in proper places
Not using the square root properly is another mistake that students make. To resolve this, we need to teach students 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 students 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 the 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 238 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √238?
The area of the square is approximately 238 square units.
Explanation
The area of the square = side^2.
The side length is given as √238.
Area of the square = side^2 = √238 × √238 = 238.
Therefore, the area of the square box is approximately 238 square units.
Problem 2
A square-shaped building measuring 238 square feet is built. If each of the sides is √238, what will be the square feet of half of the building?
119 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 238 by 2 = we get 119.
So half of the building measures 119 square feet.
Problem 3
Calculate √238 x 5.
Approximately 77.135
Explanation
The first step is to find the square root of 238, which is approximately 15.4272.
The second step is to multiply 15.4272 with 5.
So 15.4272 x 5 ≈ 77.135.
Problem 4
What will be the square root of (232 + 6)?
The square root is 16.
Explanation
To find the square root, we need to find the sum of (232 + 6).
232 + 6 = 238, and then √238 ≈ 15.42.
Therefore, the square root of (232 + 6) is approximately ±15.42.
Problem 5
Find the perimeter of the rectangle if its length ‘l’ is √238 units and the width ‘w’ is 38 units.
We find the perimeter of the rectangle as approximately 106.854 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√238 + 38).
Perimeter = 2 × (15.4272 + 38)
≈ 2 × 53.4272
≈ 106.854 units.
FAQ on Square Root of 238
1.What is √238 in its simplest form?
2.Mention the factors of 238.
3.Calculate the square of 238.
4.Is 238 a prime number?
5.238 is divisible by?
6.How does learning Algebra help students in Canada make better decisions in daily life?
7.How can cultural or local activities in Canada support learning Algebra topics such as Square Root of 238?
8.How do technology and digital tools in Canada support learning Algebra and Square Root of 238?
9.Does learning Algebra support future career opportunities for students in Canada?
Important Glossaries for the Square Root of 238
- 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 a principal square root.
- Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
- Perfect square: A perfect square is a number that is the square of an integer. For example, 4, 9, 16, and 25 are perfect squares.
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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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