%5E200.jpeg "(1+i/1-i)^200")
Exploring the Power of Complex Numbers: (1 + i)/(1 - i) ^ 200
This article will delve into the fascinating world of complex numbers by exploring the expression (1 + i)/(1 - i) ^ 200. We'll examine its simplification and the pattern that emerges when dealing with such high powers.
Simplifying the Base:
Before tackling the power of 200, let's simplify the base of the expression:
(1 + i)/(1 - i)
To simplify, we multiply both the numerator and denominator by the conjugate of the denominator (1 + i):
(1 + i)/(1 - i) * (1 + i)/(1 + i) = (1 + 2i + i^2)/(1 - i^2)
Since i^2 = -1, we can further simplify:
(1 + 2i - 1)/(1 + 1) = 2i/2 = i
Therefore, the base of our expression is simply i.
The Power of i:
Now, we need to find i ^ 200. Let's look at the pattern of powers of i:
- i^1 = i
- i^2 = -1
- i^3 = -i
- i^4 = 1
Notice that the pattern repeats every four powers. We can use this to find i ^ 200:
- 200 divided by 4 equals 50, meaning i^200 will repeat the pattern 50 times.
- Since the cycle ends with i^4 = 1, i ^ 200 = 1.
Final Result:
Therefore, (1 + i)/(1 - i) ^ 200 = i ^ 200 = 1.
Conclusion:
Through careful manipulation and understanding the pattern of powers of i, we were able to simplify a complex expression with a high power. This exercise highlights the elegance and power of complex numbers and how they can be used to solve seemingly challenging problems.