%5E6%2B(1-i)%5E3.jpeg "(1+i)^6+(1-i)^3")
Exploring the Complex Expression: (1 + i)^6 + (1 - i)^3
This article will delve into the process of simplifying the complex expression (1 + i)^6 + (1 - i)^3, showcasing the key concepts and techniques involved.
Understanding Complex Numbers and De Moivre's Theorem
Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit, defined as the square root of -1. To simplify the given expression, we'll employ De Moivre's Theorem, a fundamental theorem in complex analysis. It states:
[cos(θ) + i sin(θ)]^n = cos(nθ) + i sin(nθ)
This theorem allows us to raise complex numbers in polar form to any power conveniently.
Simplifying (1 + i)^6
- Polar Form:
We need to convert (1 + i) into polar form. The magnitude of (1 + i) is √(1^2 + 1^2) = √2, and the angle (argument) is arctan(1/1) = 45°. Therefore, (1 + i) = √2(cos(45°) + i sin(45°)). - Applying De Moivre's Theorem: Using De Moivre's theorem, (1 + i)^6 = (√2)^6(cos(645°) + i sin(645°)) = 8(cos(270°) + i sin(270°)).
- Simplifying: We know that cos(270°) = 0 and sin(270°) = -1. So, (1 + i)^6 = 8(0 + i(-1)) = -8i.
Simplifying (1 - i)^3
- Polar Form: Similar to the previous step, (1 - i) = √2(cos(-45°) + i sin(-45°)).
- Applying De Moivre's Theorem: (1 - i)^3 = (√2)^3(cos(3*(-45°)) + i sin(3*(-45°))) = 2√2(cos(-135°) + i sin(-135°)).
- Simplifying: cos(-135°) = -√2/2 and sin(-135°) = -√2/2. Therefore, (1 - i)^3 = 2√2(-√2/2 + i(-√2/2)) = -2 - 2i.
Combining the Results
Finally, to find (1 + i)^6 + (1 - i)^3, we add the simplified expressions we obtained:
(1 + i)^6 + (1 - i)^3 = -8i + (-2 - 2i) = -2 - 10i
Conclusion
By skillfully applying De Moivre's theorem and simplifying the resulting trigonometric expressions, we successfully reduced the complex expression (1 + i)^6 + (1 - i)^3 to its simplest form: -2 - 10i. This exercise highlights the power of complex numbers and their representation in polar form for simplifying calculations involving powers of complex numbers.