(x^y)^z = 64

		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!!! =) Melissa
	Hi Melissa, First find the prime factorization of 64. 64 = 2⁶. Thus you want all x, y and z so that (x^y)^z = 2⁶ Since the right side is a power of 2, the left side must also be a power of 2. Thus x^y is a power of 2. Again, since x^y 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. Cheers, Penny
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