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. The square root has applications in fields such as vehicle design, finance, and more. Here, we will discuss the square root of 6084.
The square root is the inverse operation of squaring a number. 6084 is a perfect square. The square root of 6084 can be expressed in both radical and exponential forms. In radical form, it is expressed as √6084, and in exponential form, it is (6084)^(1/2). The square root of 6084 is 78, which is a rational number because it can be expressed as a fraction where both the numerator and the denominator are integers.
The prime factorization method is suitable for perfect square numbers. For non-perfect squares, other methods like the long-division and approximation methods are used. Let's explore these methods:
The prime factorization of a number involves expressing it as a product of its prime factors. Let's see how 6084 is broken down into its prime factors.
Step 1: Finding the prime factors of 6084. Breaking it down, we get 2 × 2 × 3 × 3 × 13 × 13.
Step 2: We form pairs of prime factors. The pairs are (2, 2), (3, 3), and (13, 13). Step 3: Taking one number from each pair, we get 2 × 3 × 13 = 78.
Therefore, √6084 = 78.
The long division method is particularly useful for non-perfect square numbers, but it can be used for perfect squares as well. Let's find the square root of 6084 using the long division method, step by step.
Step 1: Pair the digits of 6084 from right to left, giving us (60)(84).
Step 2: Find the largest number whose square is less than or equal to 60. This number is 7, because 7 × 7 = 49. Subtract 49 from 60, leaving a remainder of 11.
Step 3: Bring down the next pair of digits, 84, to get 1184.
Step 4: Double the quotient obtained (7), giving 14, and use it as the starting point for the new divisor.
Step 5: Find a digit n such that 14n × n is less than or equal to 1184. The number is 8, since 148 × 8 = 1184.
Step 6: Subtract 1184 from 1184, resulting in a remainder of 0.
Thus, the quotient is 78, so the square root of 6084 is 78.
The approximation method is useful for estimating the square root of a number. Here's how to find the square root of 6084 using this method.
Step 1: Identify the closest perfect squares around 6084.
The nearest perfect squares are 5776 (76²) and 6400 (80²). √6084 lies between 76 and 80.
Step 2: Since 6084 is a perfect square, we can use the long division or prime factorization method to find the exact square root, which is 78.
Thus, using approximation, √6084 ≈ 78.
Students often make mistakes when finding square roots, such as neglecting the negative square root or misapplying the long division method. Let's explore these mistakes in detail.
Can you help Max find the area of a square if its side length is √6084?
The area of the square is 6084 square units.
The area of a square is given by side².
The side length is √6084, which is 78.
Area = side² = 78 × 78 = 6084.
Therefore, the area of the square is 6084 square units.
A square-shaped garden measures 6084 square feet. What is the length of each side?
Each side of the garden measures 78 feet.
To find the side length of the square, calculate the square root of the area. √6084 = 78.
Thus, each side of the garden is 78 feet long.
Calculate √6084 × 5.
390
First, find the square root of 6084, which is 78.
Then multiply 78 by 5. 78 × 5 = 390.
So, √6084 × 5 = 390.
What will be the square root of (6084 + 16)?
The square root is 80.
First, find the sum of 6084 and 16, which is 6100.
Next, find the square root of 6100.
Since 6100 is not a perfect square, use approximation: it is close to 80².
Thus, √6100 is approximately 80.
Find the perimeter of a rectangle if its length 'l' is √6084 units and the width 'w' is 38 units.
The perimeter of the rectangle is 232 units.
Perimeter of a rectangle = 2 × (length + width).
Length = √6084 = 78 units.
Perimeter = 2 × (78 + 38) = 2 × 116 = 232 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.