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 the field of vehicle design, finance, etc. Here, we will discuss the square root of 3024.
The square root is the inverse of the square of the number. 3024 is not a perfect square. The square root of 3024 is expressed in both radical and exponential form. In the radical form, it is expressed as √3024, whereas (3024)^(1/2) in the exponential form. √3024 ≈ 54.9808, 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, 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 3024 is broken down into its prime factors.
Step 1: Finding the prime factors of 3024 Breaking it down, we get 2 × 2 × 2 × 2 × 3 × 3 × 3 × 7: 2^4 × 3^3 × 7^1
Step 2: Now we found the prime factors of 3024. The second step is to make pairs of those prime factors. Since 3024 is not a perfect square, calculating its square root using prime factorization alone is not straightforward. We can simplify √3024 as 2^2 × 3 × √(3 × 7) = 12 × √21, but this doesn't yield an exact integer result.
The long division method is particularly used for non-perfect square numbers. This method involves a series of steps to approximate the square root value.
Step 1: To begin with, we need to group the numbers from right to left. In the case of 3024, we group it as 24 and 30.
Step 2: Now we need to find n whose square is less than or equal to 30. We can say n is 5 because 5 × 5 = 25, which is less than 30. Now the quotient is 5, and after subtracting 25 from 30, the remainder is 5.
Step 3: Bring down 24, making the new dividend 524. Add the old divisor with the same number, 5 + 5 = 10, which will be our new divisor.
Step 4: We find a number n such that 10n × n ≤ 524. Let n be 5; 105 × 5 = 525, which is greater than 524, so n should be 4. Hence, 104 × 4 = 416.
Step 5: Subtract 416 from 524, resulting in 108. The quotient is now 54.
Step 6: Since the dividend is less than the divisor, add a decimal point and two zeros to the dividend, making it 10800.
Step 7: The new divisor becomes 108. Find a number n such that 108n × n ≤ 10800. Continue this process to determine the decimal places. So the square root of √3024 is approximately 54.98.
The approximation method is another method for finding square roots. It is an easy method to estimate the square root of a given number.
Step 1: Find the closest perfect squares around 3024. The smallest perfect square less than 3024 is 2916 (54^2), and the largest perfect square greater than 3024 is 3136 (56^2). Thus, √3024 is between 54 and 56.
Step 2: Use interpolation: (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square) = (3024 - 2916) / (3136 - 2916) = 108 / 220 ≈ 0.49. So the square root of 3024 is approximately 54 + 0.49 = 54.49.
Students do 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 √3024?
The area of the square is approximately 3024 square units.
The area of the square = side^2.
The side length is given as √3024.
Area of the square = side^2 = √3024 × √3024 = 3024.
Therefore, the area of the square box is approximately 3024 square units.
A square-shaped building measuring 3024 square feet is built; if each of the sides is √3024, what will be the square feet of half of the building?
1512 square feet
We can divide the given area by 2 since the building is square-shaped.
Dividing 3024 by 2 = 1512.
So half of the building measures 1512 square feet.
Calculate √3024 × 5.
Approximately 274.9
The first step is to find the square root of 3024, which is approximately 54.98.
The second step is to multiply 54.98 by 5.
So 54.98 × 5 ≈ 274.9.
What will be the square root of (3024 + 16)?
The square root is approximately 55.2
To find the square root, we need to find the sum of (3024 + 16) = 3040. The square root of 3040 is approximately 55.2.
Find the perimeter of the rectangle if its length ‘l’ is √3024 units and the width ‘w’ is 38 units.
The perimeter of the rectangle is approximately 185.96 units.
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√3024 + 38) = 2 × (54.98 + 38) = 2 × 92.98 ≈ 185.96 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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