But of course! Let z = 4 and let's find the "2th root." Then . (The proof of this requires the next definition in the series, so technically I suppose you can't say this yet. That will be up to your instructor.)

Hmmmm... Not true this time. Consider z = 12, a = 2, b = 3, c = 1, d = 6. Note that ab = cd = 6. But

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Did you possibly mean that given "ad = bc" that ? This one is almost true. We have to add one more definition:

Given a positive number z and a rational number q, then the "principle value" of is also positive. (If we speak of principle values, then the answer to part a) is that we can only have one principle value to .)

So if we are speaking of principle values then we have and have to prove that implies that , which is a simple proof that I leave to you.

-Dan