Last updated on May 26th, 2025
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.
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.
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:
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.
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.
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.
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.
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.
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.
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
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.
Calculate √323 x 5.
89.861
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
What will be the square root of (300 + 23)?
The square root is approximately 17.972
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.
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.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√323 + 50) = 2 × (17.972 + 50) = 2 × 67.972 = 135.9444 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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