If x is an integer, is 9^x + 9^(-x) = b?

(1) 3^x + 3^(-x) = sqrt(b+2)
(2) x>0

Is there a better way to rephrase the statement so that you aren't forced to go through the whole FOIL process in statement (1)? Just trying to find some shortcuts on the more difficult problems :)

There's a logical way to know that you don't have to go through the whole FOIL process, if you think about how the question is constructed.

If we square statement 1 (but don't actually do the math) we get:
[3^x + 3^(-x)]^2 = b+2 now move the two to the left side to get:
[3^x + 3^(-x)]^2 + 2 = b

We're asked a yes/no question: is 9^x + 9^(-x) = b?
Without actually doing the math on the left side of my statement 1 above, I know that I either can or cannot manipulate it to equal 9^x + 9^(-x). That is, I can either answer the question yes, definitively, or no, definitively. It doesn't matter whether the answer is yes or no - both are sufficient. Only if I answer the question "maybe" is it insufficient - but I know I'll be able to answer this one way or the other.

So you can stop there if you are comfortable with that logic. At the same time, know how to do the FOIL. If you learn all of your exponent rules and don't let yourself get confused b/c the terms are unusual, you should be able to do this (though, yes, it is a bit annoying).

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Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT