125 LearnersLast updated on May 26th, 2025
Square Root of 323
If a number is multiplied by itself, the result is a square. The inverse operation is finding the square root. Square roots are used in various fields such as physics, engineering, and finance. Here, we will discuss the square root of 323.
What is the Square Root of 323?
The square root is the inverse operation of squaring a number. 323 is not a perfect square. The square root of 323 can be expressed in both radical and exponential forms. In radical form, it is expressed as √323, whereas in exponential form, it is (323)^(1/2). The value of √323 is approximately 17.9722, which is an irrational number because it cannot be expressed as a fraction p/q, where p and q are integers and q ≠ 0.
Finding the Square Root of 323
The prime factorization method is used for perfect square numbers. However, for non-perfect square numbers like 323, methods such as 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 323 by Prime Factorization Method
The product of prime factors is the prime factorization of a number. Now let us look at how 323 is broken down into its prime factors:
Step 1: Finding the prime factors of 323 Breaking it down, we get 17 x 19: 17^1 x 19^1
Step 2: Now that we have found the prime factors of 323, the next step is to check for pairs of those prime factors. Since 323 is not a perfect square, the digits cannot be grouped into pairs.
Therefore, calculating √323 using prime factorization alone is not feasible.
Square Root of 323 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 numbers 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 323, we group it as 23 and 3.
Step 2: Now we need to find n whose square is closest to or equal to 3. We can say n is ‘1’ because 1 x 1 is lesser than or equal to 3. Now the quotient is 1, and after subtracting 1 from 3, the remainder is 2.
Step 3: Now let us bring down 23, which becomes the new dividend. Add the old divisor with the same number 1 + 1, 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 have 2n as the new divisor. We need to find the value of n.
Step 5: The next step is finding 2n x n ≤ 223. Let us consider n as 8; now 28 x 8 = 224.
Step 6: Since 28 x 8 = 224 is greater than 223, check 27 x 7 = 189. Subtract 189 from 223; the difference is 34, and the quotient is 17.
Step 7: Since the remainder is less than the divisor, add a decimal point, allowing us to append two zeroes to the dividend. Now the new dividend is 3400.
Step 8: Find the new divisor, which is 179 because 179 x 9 = 1611.
Step 9: Subtracting 1611 from 3400 gives the result 1789.
Step 10: Now the quotient is approximately 17.9.
Step 11: Continue these steps until reaching a desired level of precision or until the remainder is zero.
So the square root of √323 is approximately 17.972.
Square Root of 323 by Approximation Method
The approximation method is another way to find square roots. It is a straightforward method to estimate the square root of a given number. Now let us learn how to find the square root of 323 using the approximation method.
Step 1: Identify the closest perfect squares to √323.
The smallest perfect square less than 323 is 289, and the largest perfect square greater than 323 is 324. √323 falls between 17 and 18.
Step 2: Now apply the formula (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square).
Using the formula (323 - 289) / (324 - 289) = 34 / 35 = 0.9714 Adding this to 17 gives 17 + 0.9714 = 17.9714, so the square root of 323 is approximately 17.9714.
Common Mistakes and How to Avoid Them in the Square Root of 323
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's explore some 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 generally consider only the positive square root, as it is often the required one.
For example, √50 = 7.07, but there is also -7.07, which should not be overlooked.
Mistake 2
Not adding the square root symbol in proper places
Misusing the square root symbol is another common mistake. To resolve this, we need to teach students to simplify the numbers inside the square root first, then proceed to the next steps.
For example, √(12 + 24) = √36, not √12 + √24.
Mistake 3
Not finding the correct value of a non-perfect square
The initial step is to simplify the numbers inside the square root to avoid mistakes while identifying the approximate value.
For example, √50 = 7 is incorrect; the correct answer is √50 = 7.071067.
Mistake 4
Confusing the square root symbol with the cube root
Students need constant reminders about the differences between cube roots and square roots to avoid confusion.
For example, √50 and ∛50 are different.
Mistake 5
Making mistakes in the long division
Finding a square root using the long division method involves several steps, and students might accidentally skip a step, leading to incorrect results. This can happen during subtraction or at other points in the process.
Square root of 323 Examples
Problem 1
Can you help Max find the area of a square box if its side length is given as √323?
The area of the square is 1041.216 square units.
Explanation
The area of the square = side².
The side length is given as √323.
Area of the square = side² = √323 x √323 = 17.9722 × 17.9722 = 1041.216
Therefore, the area of the square box is 1041.216 square units.
Problem 2
A square-shaped building measuring 323 square feet is built; if each of the sides is √323, what will be the square feet of half of the building?
161.5 square feet
Explanation
We can just divide the given area by 2 as the building is square-shaped.
Dividing 323 by 2 = 161.5
So half of the building measures 161.5 square feet.
Problem 3
Calculate √323 x 5.
89.861
Explanation
The first step is to find the square root of 323, which is approximately 17.972, then multiply 17.972 by 5. So 17.972 x 5 = 89.861
Problem 4
What will be the square root of (300 + 23)?
The square root is approximately 17.972
Explanation
To find the square root, calculate the sum of (300 + 23). 300 + 23 = 323, and √323 is approximately 17.972.
Therefore, the square root of (300 + 23) is approximately ±17.972.
Problem 5
Find the perimeter of a rectangle if its length ‘l’ is √323 units and the width ‘w’ is 50 units.
The perimeter of the rectangle is approximately 135.9444 units.
Explanation
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√323 + 50) = 2 × (17.972 + 50) = 2 × 67.972 = 135.9444 units.
FAQ on Square Root of 323
1.What is √323 in its simplest form?
2.Mention the factors of 323.
3.Calculate the square of 323.
4.Is 323 a prime number?
5.What is 323 divisible by?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Square Root of 323?
8.How do technology and digital tools in Australia support learning Algebra and Square Root of 323?
9.Does learning Algebra support future career opportunities for students in Australia?
Important Glossaries for the Square Root of 323
- Square root: A square root is the inverse operation of squaring a number. Example: 4^2 = 16, and the inverse is the square root, √16 = 4.
- Irrational number: An irrational number is a number that cannot be written in the form p/q, where q ≠ 0, and p and q are integers.
- Principal square root: A number has both positive and negative square roots, but the positive square root is more commonly used, known as the principal square root.
- Prime factorization: The expression of a number as a product of prime numbers. Example: 323 = 17 x 19.
- Approximation: Estimating a value that is close to the actual value, often used for non-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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