Quandaries and Queries


My name is Melissa and I am taking a college course for elementary geometry teachers. I have come acrossed a problem that I am having problems with. The question is:

How many ordered triples of positive intergers (xyz) satisfy (x[raised to the y])raised to the z = 64?

~raised to the means squared, tripled, etc.~

Any help you can give me would be greatly appreciated! Thank you so much!!! =)




Hi Melissa,

First find the prime factorization of 64. 64 = 26. Thus you want all x, y and z so that

(xy)z = 26 Since the right side is a power of 2, the left side must also be a power of 2. Thus xy is a power of 2. Again, since xy is a power of 2, x must be a power of 2. In fact x is 21, or 22,..., or 26. Try these possible values for x and see what y and z can be.


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