Math Problem Statement
Solution
The given expression in the image is:
This expression involves complex numbers due to the square root of , which can be simplified using imaginary numbers (), where . Thus, .
Rewriting the expression:
Next, we can recognize that the terms inside the parentheses resemble the cube roots of unity. These cube roots are related to the complex numbers and , where:
Thus, the expression can be interpreted as involving powers of the cube roots of unity.
Cube roots of unity properties:
Since both powers of and are integers that are multiples of 3, the expression simplifies as follows:
Thus, the entire expression becomes:
The simplified result is:
Would you like more details or have any questions? Here are five related questions you can explore:
- What are the cube roots of unity, and how are they used in complex number calculations?
- How do imaginary numbers like and work in mathematical expressions?
- Can we generalize this result for other powers of complex roots?
- What is the geometric interpretation of complex numbers and roots of unity on the complex plane?
- How do De Moivre’s theorem and Euler’s formula relate to powers of complex numbers?
Tip: Understanding the cube roots of unity helps in simplifying many complex expressions, especially those involving periodicity.
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Math Problem Analysis
Mathematical Concepts
Complex Numbers
Roots of Unity
Imaginary Numbers
Formulas
(√-3) = i√3
Cube roots of unity: ω = (1 + i√3)/2, ω^2 = (1 - i√3)/2
ω^3 = 1
Theorems
Properties of Cube Roots of Unity
Powers of Complex Numbers
De Moivre's Theorem
Suitable Grade Level
Grades 11-12
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