Last updated on May 26th, 2025
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.
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.
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:
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.
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.
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.
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.
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.
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.
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
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.
Calculate √231 x 5.
Approximately 75.99
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.
What will be the square root of (225 + 6)?
The square root is approximately 15.81.
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.
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.
Perimeter of a rectangle = 2 × (length + width)
Perimeter = 2 × (√231 + 38)
= 2 × (15.20 + 38)
= 2 × 53.20
= 106.40 units.
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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