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Simplifying Complex Expressions: (1+i)^3 + (1-i)^6
This article will explore the simplification of the complex expression (1+i)^3 + (1-i)^6. We'll utilize the properties of complex numbers and binomial expansion to arrive at a concise solution.
Understanding the Problem
We are dealing with complex numbers in the form of a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (i² = -1). Our task is to calculate the sum of two complex expressions raised to powers.
Simplifying using Binomial Expansion
Let's break down the problem into manageable parts:
1. Expanding (1+i)^3:
We can use the binomial theorem to expand this expression:
(1+i)³ = 1³ + 3(1²)(i) + 3(1)(i²) + i³
Remember that i² = -1 and i³ = i² * i = -i. Substituting these values, we get:
(1+i)³ = 1 + 3i - 3 - i = -2 + 2i
2. Expanding (1-i)^6:
Similarly, we can expand (1-i)^6 using the binomial theorem:
(1-i)⁶ = 1⁶ + 6(1⁵)(-i) + 15(1⁴)(-i)² + 20(1³)(-i)³ + 15(1²)(-i)⁴ + 6(1)(-i)⁵ + (-i)⁶
Simplifying using the properties of 'i', we get:
(1-i)⁶ = 1 - 6i - 15 + 20i + 15 - 6i - 1 = -2 + 8i
Final Calculation
Now we have the simplified forms of both expressions:
(1+i)³ = -2 + 2i (1-i)⁶ = -2 + 8i
Adding these together:
(-2 + 2i) + (-2 + 8i) = -4 + 10i
Conclusion
Therefore, the simplified form of the expression (1+i)^3 + (1-i)^6 is -4 + 10i. By utilizing the properties of complex numbers and binomial expansion, we effectively broke down the problem and arrived at the solution.