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Simplifying the Expression: (1+i)^5(1+1/i)^5
This problem involves complex numbers and utilizes the properties of exponents and complex arithmetic. Let's break it down step by step:
Understanding Complex Numbers
- Complex numbers are numbers of the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, defined as √-1.
- Operations with complex numbers:
- Addition/Subtraction: Combine the real and imaginary parts separately.
- Multiplication: Use the distributive property and remember that i² = -1.
- Division: Multiply both numerator and denominator by the complex conjugate of the denominator.
Simplifying (1 + i)^5 and (1 + 1/i)^5
1. (1 + i)^5
- Binomial Theorem: We can use the binomial theorem to expand this. However, it's computationally easier to observe a pattern:
- (1 + i)² = 1 + 2i + i² = 2i
- (1 + i)³ = (1 + i)² (1 + i) = 2i (1 + i) = 2i + 2i² = -2 + 2i
- (1 + i)⁴ = (1 + i)³ (1 + i) = (-2 + 2i)(1 + i) = -2 - 2i + 2i + 2i² = -4
- (1 + i)⁵ = (1 + i)⁴ (1 + i) = -4 (1 + i) = -4 - 4i
2. (1 + 1/i)^5
- Simplify 1/i: Multiply numerator and denominator by i: (1/i) * (i/i) = i/i² = -i
- Rewrite the expression: (1 + 1/i)⁵ = (1 - i)⁵
- Use the Binomial Theorem or observe the pattern as done above.
3. Combining the Results
- After simplifying (1 + i)⁵ and (1 - i)⁵, we will have expressions in the form of (a + bi) and (c + di).
- Multiply these two complex numbers using the distributive property and remember i² = -1.
Final Result
The final answer will be in the form of a + bi, where 'a' and 'b' are real numbers. This simplified expression represents the result of the given complex number calculation.
Remember: While you can use the Binomial Theorem, identifying patterns in complex number powers can be quicker and less prone to calculation errors.