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 concept of square roots extends into the realm of complex numbers when dealing with negative numbers. Here, we will discuss the square root of -108.
The square root is the inverse of the square of a number. When dealing with negative numbers, we use complex numbers to express the square root. The square root of -108 is expressed in terms of the imaginary unit 'i', where i = √(-1). Thus, the square root of -108 is written as √(-108) = √(108) * i. The value of √108 is approximately 10.3923, so √(-108) ≈ 10.3923i.
There are different methods to find square roots, but for negative numbers, the concept of imaginary numbers is used. Here's how to find the square root of -108:
1. Calculate the square root of the positive part, 108.
2. Multiply the result by i (the imaginary unit).
To find the square root of -108 using imaginary numbers, we start with the positive part:
Step 1: Calculate the square root of 108. The prime factorization of 108 is 2 x 2 x 3 x 3 x 3, or 2² x 3³.
Step 2: Simplify √108 = √(2² x 3² x 3) = 2 x 3 x √3 = 6√3 ≈ 10.3923.
Step 3: Multiply by the imaginary unit i: √(-108) = √108 * i = 10.3923i.
The long division method is typically used for approximating square roots of positive numbers. For -108, we focus on the positive part:
Step 1: Use long division to approximate √108, which we already found to be about 10.3923.
Step 2: Multiply this result by i to obtain the square root of the negative number: √(-108) = 10.3923i.
Approximation helps find the square root of the positive part, 108:
Step 1: Identify the perfect squares surrounding 108. 100 (10²) and 121 (11²) are the closest perfect squares.
Step 2: Approximate between these values: √108 is between 10 and 11. Using the approximation method, we find √108 ≈ 10.3923.
Step 3: Multiply by i for the imaginary part: √(-108) = 10.3923i.
Students often make mistakes when dealing with the square roots of negative numbers, especially with the involvement of imaginary numbers. Let's discuss some common mistakes and how to avoid them.
Can you help Max find the area of a square box if its side length is given as √(-108)?
The area of the square is -108 square units.
The area of the square = side².
The side length is given as √(-108).
Area of the square = (√(-108))² = -108.
Therefore, the area of the square box is -108 square units, considering complex units.
A complex square-shaped plot has an area of -108 square meters. What is the length of each side if it's given by √(-108)?
The side length is approximately 10.3923i meters.
The side length of the square is given by the square root of the area. √(-108) = 10.3923i meters, which is the length of each side, considering the imaginary component.
Calculate √(-108) x 5.
51.9615i
First, find the square root of -108, which is approximately 10.3923i. Multiply by 5: 10.3923i x 5 = 51.9615i.
What will be the square root of (-100 + 8)?
The square root is approximately 10.3923i.
Calculate the sum: (-100 + 8) = -92. Then find the square root: √(-92) = √92 * i ≈ 9.5917i.
Find the perimeter of a rectangle if its length 'l' is √(-108) units and the width 'w' is 38 units.
The perimeter is a complex number: 76 + 20.7846i units.
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√(-108) + 38) = 2 × (10.3923i + 38). = 76 + 20.7846i 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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