159 LearnersLast updated on May 26th, 2025
Square Root of -108
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.
What is 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.
Finding the Square Root of -108
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).
Square Root of -108 Using Imaginary Numbers
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.
Square Root of -108 by Long Division Method (Imaginary Part)
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.
Square Root of -108 by Approximation Method
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.
Common Mistakes and How to Avoid Them in the Square Root of -108
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.
Mistake 1
Ignoring the Imaginary Unit
When dealing with negative square roots, it's crucial to remember the imaginary unit 'i'.
For example, √(-50) = √50 * i = 7.07i, not just 7.07.
Mistake 2
Incorrect Simplification of Radicals
Simplifying the square root of a number involves correctly identifying and pairing factors.
For example, √(12+24) = √36 = 6, not √12 + √24.
Mistake 3
Not Recognizing the Need for Complex Numbers
For negative square roots, it's essential to recognize the need for complex numbers.
For example, without recognizing the negative sign's impact, √(-50) might be incorrectly simplified to 7.07 instead of 7.07i.
Mistake 4
Confusing Square Roots and Cube Roots
Students sometimes confuse square root and cube root symbols, leading to incorrect calculations.
For example, √50 and ∛50 are different.
Mistake 5
Errors in Approximation
Approximating square roots requires careful calculation. Mistakes in these approximations can lead to significant errors in the final answer.
Square Root of -108 Examples
Problem 1
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.
Explanation
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.
Problem 2
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.
Explanation
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.
Problem 3
Calculate √(-108) x 5.
51.9615i
Explanation
First, find the square root of -108, which is approximately 10.3923i. Multiply by 5: 10.3923i x 5 = 51.9615i.
Problem 4
What will be the square root of (-100 + 8)?
The square root is approximately 10.3923i.
Explanation
Calculate the sum: (-100 + 8) = -92. Then find the square root: √(-92) = √92 * i ≈ 9.5917i.
Problem 5
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.
Explanation
Perimeter of the rectangle = 2 × (length + width). Perimeter = 2 × (√(-108) + 38) = 2 × (10.3923i + 38). = 76 + 20.7846i units.
FAQ on Square Root of -108
1.What is √(-108) in its simplest form?
2.What is the principal square root of -108?
3.Can the square root of -108 be simplified further?
4.Is -108 a perfect square?
5.What does the imaginary unit 'i' represent?
6.How does learning Algebra help students in India make better decisions in daily life?
7.How can cultural or local activities in India support learning Algebra topics such as Square Root of -108?
8.How do technology and digital tools in India support learning Algebra and Square Root of -108?
9.Does learning Algebra support future career opportunities for students in India?
Important Glossaries for the Square Root of -108
- Imaginary Number: A number that can be written as a real number multiplied by the imaginary unit 'i', which is defined by i² = -1.
- Complex Number: A number that has both a real part and an imaginary part, expressed in the form a + bi, where a and b are real numbers.
- Square Root: The value that, when multiplied by itself, gives the original number. For negative numbers, it involves the imaginary unit.
- Perfect Square: A number that is the square of an integer. Negative numbers cannot be perfect squares.
- Approximation: The process of finding a number close to the exact value, often used for irrational numbers or complex calculations.
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Jaskaran Singh Saluja
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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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