# Thread: Explain why (3+2i) raised to any positive exponent will remain a complex number.

1. ## Explain why (3+2i) raised to any positive exponent will remain a complex number.

The question asks, "Explain why (3+2i) raised to any positive exponent will remain a complex number"
I understand the math behind why it will remain a complex number, I am just struggling to piece it into words. If you can offer a clear explanation it will be helpful.

2. Originally Posted by evking
The question asks, "Explain why (3+2i) raised to any positive exponent will remain a complex number"
I understand the math behind why it will remain a complex number, I am just struggling to piece it into words. If you can offer a clear explanation it will be helpful.
Any answer you come up with is going to be rather mathy.

The easiest but defective answer I can give is that the only way you can multiply two complex numbers and get a real product is to multiply a complex number by its conjugate and no complex number is its own conjugate. To make that answer complete, however, you would need to prove that no positive power of a complex number equals its conjugate.

That is not nearly the best answer though. The best answer is that the product of two complex numbers is always a complex number because the real numbers are a subset of the complex numbers.

3. Originally Posted by evking
The question asks, "Explain why (3+2i) raised to any positive exponent will remain a complex number"
I understand the math behind why it will remain a complex number, I am just struggling to piece it into words. If you can offer a clear explanation it will be helpful.
Can you tell us something about the context of the question, so we can have a better idea what sort of explanation is expected? What course is it, what has been taught so far, have there been proofs and theorems, and so on.

Basically, as I understand it, you are just expected to talk about how raising to a positive integer power means multiplying complex numbers together, and why that will always result in a complex number (that is, you will have a real part and an imaginary part, either of which might be zero, but nothing else). Have you learned the phrase "closed under multiplication"?

If you can just tell us what ideas you have, based on what you have been taught, someone may be able to help you word it better. But I don't think you really need any fancy wording; the word "proof" isn't used. And the question itself is vague, so your answer doesn't have to be very precise either.

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