119 LearnersLast updated on May 26th, 2025
Square Root of 233
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 233.
What is the Square Root of 233?
The square root is the inverse of the square of the number. 233 is not a perfect square. The square root of 233 is expressed in both radical and exponential forms. In the radical form, it is expressed as √233, whereas (233)^(1/2) in the exponential form. √233 ≈ 15.2643, 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 233
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 233 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 233 is broken down into its prime factors.
Step 1: Finding the prime factors of 233 233 is a prime number, so it cannot be broken down into smaller prime factors.
Step 2: Since 233 is a prime number and not a perfect square, calculating the square root using prime factorization is impossible.
Square Root of 233 by Long Division Method
The long division method is particularly used for non-perfect square numbers. Let us now learn how to find the square root using the long division method, step by step.
Step 1: Group the numbers from right to left. In the case of 233, we need to group it as 33 and 2.
Step 2: 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 33, which is the new dividend. Add the old divisor (1) to itself, which gives us 2 as the new divisor.
Step 4: Use 2 as the new divisor to find a value of n such that 2n x n ≤ 133. Let us consider n as 5, now 25 x 5 = 125.
Step 5: Subtract 125 from 133, the difference is 8.
Step 6: 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 800.
Step 7: Find the new divisor, which is 105 because 1050 x 5 = 525.
Step 8: Subtracting 525 from 800 gives us the result 275.
Step 9: Continue doing these steps until we get two numbers after the decimal point. So the square root of √233 is approximately 15.26.
Square Root of 233 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 233 using the approximation method.
Step 1: Find the closest perfect square of √233. The smallest perfect square less than 233 is 225, and the largest perfect square greater than 233 is 256. √233 falls somewhere between 15 and 16.
Step 2: Apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Using the formula (233 - 225) / (256 - 225) = 0.258. Adding the integer part to the decimal, 15 + 0.258 ≈ 15.26.
Common Mistakes and How to Avoid Them in the Square Root of 233
Students often make mistakes while finding the square root, such as forgetting about the negative square root or skipping 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 to remind students that a number has both positive and negative square roots. However, we usually consider only the positive square root, as it is the one required in most contexts.
For example, √50 = 7.07, but there is also -7.07, which should not be forgotten.
Mistake 2
Not adding the square root symbol in the proper places
Not using the square root symbol properly is another mistake students make. Teach students to simplify the numbers inside the square root before proceeding.
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
Students should be taught to simplify numbers inside the square root to avoid mistakes in identifying approximate values.
For example, √50 ≠ 7; the correct answer is √50 = 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Teach students the difference between cube root and square root to prevent confusion.
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. Students may accidentally skip a step, leading to incorrect results. This may occur during subtraction or other steps.
Square root of 233 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √233?
The area of the square is 233 square units.
Explanation
The area of the square = side².
The side length is given as √233.
Area of the square = side² = √233 x √233 = 233.
Therefore, the area of the square box is 233 square units.
Problem 2
A square-shaped building measuring 233 square feet is built; if each of the sides is √233, what will be the square feet of half of the building?
116.5 square feet
Explanation
We can divide the given area by 2 as the building is square-shaped.
Dividing 233 by 2 gives us 116.5.
So half of the building measures 116.5 square feet.
Problem 3
Calculate √233 x 5.
76.32
Explanation
The first step is to find the square root of 233, which is approximately 15.26.
The second step is to multiply 15.26 by 5.
So 15.26 x 5 = 76.32.
Problem 4
What will be the square root of (225 + 8)?
The square root is approximately 15.26.
Explanation
To find the square root, we need to find the sum of (225 + 8).
225 + 8 = 233, and then √233 ≈ 15.26.
Therefore, the square root of (225 + 8) is approximately ±15.26.
Problem 5
Find the perimeter of the rectangle if its length 'l' is √233 units and the width 'w' is 38 units.
We find the perimeter of the rectangle as approximately 106.52 units.
Explanation
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√233 + 38)
= 2 × (15.26 + 38)
= 2 × 53.26
= 106.52 units.
FAQ on Square Root of 233
1.What is √233 in its simplest form?
2.Is 233 a prime number?
3.Calculate the square of 233.
4.Is 233 a perfect square?
5.233 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 233?
8.How do technology and digital tools in India support learning Algebra and Square Root of 233?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of 233
- 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 cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
- Prime number: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
- Perfect square: A perfect square is a number that is the square of an integer.
- Long division method: A method used to find the square root of non-perfect squares, involving grouping numbers and dividing them step by step.
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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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