129 LearnersLast updated on May 26th, 2025
Square Root of 231
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 various fields such as vehicle design, finance, etc. Here, we will discuss the square root of 231.
What is the Square Root of 231?
The square root is the inverse of the square of a number. 231 is not a perfect square. The square root of 231 can be expressed in both radical and exponential forms. In the radical form, it is expressed as √231, whereas in the exponential form, it is written as (231)^(1/2). √231 ≈ 15.1987, 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 231
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 231, the long division method and approximation method are more suitable. Let us now learn these methods:
- Prime factorization method
- Long division method
- Approximation method
Square Root of 231 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 231 is broken down into its prime factors:
Step 1: Finding the prime factors of 231. Breaking it down, we get 3 x 7 x 11: 3¹ x 7¹ x 11¹
Step 2: Now we have found the prime factors of 231. Since 231 is not a perfect square, we cannot pair the digits completely. Therefore, calculating the square root of 231 using the prime factorization method is not possible.
Square Root of 231 by Long Division Method
The long division method is particularly used for non-perfect square numbers. In this method, we should check for the closest perfect square number to 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 231, we need to group it as 31 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: Bring down 31, making the new dividend 131. Add the old divisor with the same number, 1 + 1, to get 2, which will be part of our new divisor.
Step 4: The new divisor will be the sum of the previous divisor and quotient. Now we get 2n as the new divisor, and we need to find the value of n.
Step 5: Find 2n × n ≤ 131. Let us consider n as 5; now 25 x 5 = 125.
Step 6: Subtract 125 from 131, the difference is 6, and the quotient becomes 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 600.
Step 8: The new divisor is 105, because 105 x 5 = 525.
Step 9: Subtracting 525 from 600, we get the result 75.
Step 10: Now the quotient is 15.1.
Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no more decimal values, continue until the remainder is zero. So the square root of √231 is approximately 15.20.
Square Root of 231 by Approximation Method
The approximation method is another way to find 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 231 using the approximation method:
Step 1: Now we have to find the closest perfect square of √231. The smallest perfect square less than 231 is 225, and the largest perfect square greater than 231 is 256. √231 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). Using the formula: (231 - 225) / (256 - 225) = 6 / 31 ≈ 0.1935 Adding this to the nearest smaller perfect square root gives us 15 + 0.1935 ≈ 15.1935. So, the square root of 231 is approximately 15.20.
Common Mistakes and How to Avoid Them in the Square Root of 231
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping steps in the long division method. Let us look at a few common mistakes in detail.
Mistake 1
Forgetting about the negative square root
It is important for students to remember that a number has both positive and negative square roots. However, we typically use the positive square root for practical purposes.
For example: √50 = 7.07, but there is also -7.07, which should not be ignored.
Mistake 2
Not adding the square root symbol in the proper places
Not using the square root symbol correctly is another mistake that students often make. To resolve this, we need to teach students to simplify the numbers inside the square root first and then proceed to the next step.
For example: √(12 + 24) ≠ √12 + √24. The correct method 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; 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, 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. Because of this, many students might skip a step accidentally, resulting in a wrong answer. This error can occur in subtraction or elsewhere in the process.
Square Root of 231 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √231?
The area of the square is approximately 231 square units.
Explanation
The area of a square = side².
The side length is given as √231.
Area of the square = side² = √231 x √231 = 231.
Therefore, the area of the square box is approximately 231 square units.
Problem 2
A square-shaped building measuring 231 square feet is built. If each of the sides is √231, what will be the square feet of half of the building?
115.5 square feet
Explanation
We can divide the given area by 2 since the building is square-shaped.
Dividing 231 by 2 gives 115.5.
So half of the building measures 115.5 square feet.
Problem 3
Calculate √231 x 5.
Approximately 75.99
Explanation
The first step is to find the square root of 231, which is approximately 15.20.
The second step is to multiply 15.20 by 5.
So, 15.20 x 5 ≈ 76.
Problem 4
What will be the square root of (225 + 6)?
The square root is approximately 15.81.
Explanation
To find the square root, calculate the sum of (225 + 6).
225 + 6 = 231, and then √231 ≈ 15.20.
Therefore, the square root of (225 + 6) is approximately 15.20.
Problem 5
Find the perimeter of a rectangle if its length 'l' is √231 units and the width 'w' is 38 units.
The perimeter of the rectangle is approximately 106.40 units.
Explanation
Perimeter of a rectangle = 2 × (length + width)
Perimeter = 2 × (√231 + 38)
= 2 × (15.20 + 38)
= 2 × 53.20
= 106.40 units.
FAQ on Square Root of 231
1.What is √231 in its simplest form?
2.Mention the factors of 231.
3.Calculate the square of 231.
4.Is 231 a prime number?
5.231 is divisible by?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of 231?
8.How do technology and digital tools in India support learning Algebra and Square Root of 231?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 231
- Square root: A square root is the inverse of a square. Example: 4² = 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, the positive square root is more commonly used in practical applications. 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, for example, 231 = 3 x 7 x 11.
- Decimal: If a number has a whole number and a fraction in a single number, 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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