Last updated on May 26th, 2025
If a number is multiplied by itself, the result is a square. The inverse operation of finding a square is called finding the square root. The square root is used in various fields, including vehicle design, finance, etc. Here, we will discuss the square root of 339.
The square root is the inverse operation of squaring a number. 339 is not a perfect square. The square root of 339 is expressed in both radical and exponential form. In radical form, it is expressed as √339, whereas in exponential form it is expressed as (339)^(1/2). √339 ≈ 18.412, 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, 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:
The product of prime factors is the prime factorization of a number. Now let us look at how 339 is broken down into its prime factors:
Step 1: Finding the prime factors of 339 Breaking it down, we get 3 x 113
Step 2: Now we have found the prime factors of 339. Since 339 is not a perfect square, the digits of the number can’t be grouped into pairs.
Therefore, calculating √339 using prime factorization is not possible.
The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number 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 339, we need to group it as 39 and 3.
Step 2: Now we need to find n whose square is less than or equal to 3. We can say n is ‘1’ because 1 x 1 is less than or equal to 3. Now the quotient is 1; after subtracting 1 from 3, the remainder is 2.
Step 3: Now let us bring down 39, which is the new dividend. Add the old divisor with the same number 1 + 1 to get 2, which will be our new divisor.
Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor, and we need to find the value of n.
Step 5: The next step is finding 2n × n ≤ 239. Let us consider n as 8, now 28 x 8 = 224
Step 6: Subtract 224 from 239, and the difference is 15. The quotient is 18.
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 1500.
Step 8: Now we need to find the new divisor. Considering 286, because 286 x 6 = 1716 is too large, we try 285 x 5 = 1425.
Step 9: Subtracting 1425 from 1500 gives 75.
Step 10: Now the quotient is 18.4
Step 11: Continue doing these steps until we get two numbers after the decimal point. If there are no decimal values, continue until the remainder is zero.
So the square root of √339 is approximately 18.41.
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 339 using the approximation method.
Step 1: We need to find the closest perfect square to √339.
The smallest perfect square less than 339 is 324, and the largest perfect square more than 339 is 361. √339 falls somewhere between 18 and 19.
Step 2: Now we need to apply the formula: (Given number - smallest perfect square) ÷ (Greater perfect square - smallest perfect square)
Using the formula (339 - 324) ÷ (361 - 324) = 15 ÷ 37 = 0.405 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 18 + 0.405 ≈ 18.41.
So the square root of 339 is approximately 18.41.
Students often make mistakes while finding the square root, such as forgetting about the negative square root, skipping long division methods, etc. Now let us look at a few of those mistakes that students tend to make in detail.
Can you help Max find the area of a square box if its side length is given as √339?
The area of the square is 339 square units.
The area of the square = side².
The side length is given as √339.
Area of the square = side² = √339 x √339 = 339.
Therefore, the area of the square box is 339 square units.
A square-shaped building measuring 339 square feet is built; if each of the sides is √339, what will be the square feet of half of the building?
169.5 square feet
We can just divide the given area by 2 as the building is square-shaped.
Dividing 339 by 2 = we get 169.5
So half of the building measures 169.5 square feet.
Calculate √339 x 5.
Approximately 92.06
The first step is to find the square root of 339, which is approximately 18.41.
The second step is to multiply 18.41 by 5.
So, 18.41 x 5 ≈ 92.06.
What will be the square root of (339 + 5)?
The square root is approximately 18.52.
To find the square root, we need to find the sum of (339 + 5). 339 + 5 = 344, and then √344 ≈ 18.52.
Therefore, the square root of (339 + 5) is approximately ±18.52.
Find the perimeter of the rectangle if its length ‘l’ is √339 units and the width ‘w’ is 20 units.
We find the perimeter of the rectangle is approximately 76.82 units.
Perimeter of the rectangle = 2 × (length + width)
Perimeter = 2 × (√339 + 20) ≈ 2 × (18.41 + 20) ≈ 2 × 38.41 ≈ 76.82 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.
: He loves to play the quiz with kids through algebra to make kids love it.