Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
Construct xy in stages, working up from n = 1:
xy = xy
Now, to figure out xy, apply the second part of the definition, using n = 1 (odd):
xy = (xy)x = (xy)x = xyx
Be careful to apply the right part of the definition! Track carefully where n + 1 is and where n is.
Now keep going, using the third part of the definition (with n = 2, even):
xy = (xy)x = (xyx)x = xyxx
(We’ll avoid even tinier exponents by simply writing xx for x2.)
Finally, the xy expression will add another y to the exponent:
xy = (xy)y = (xyxx)y = xyyxx
Now, we are told that y is ½ and that this crazy expression xyyxx equals 2.Substituting in y = ½, you get x^(x2/4) = 2, using the “caret” (^) to indicate exponentiation.
You should recognize at this point that it’s unlikely to be most efficient to try to solve directly for x—in fact, it may not even be possible within “GMAT math.” Rather, you should glance through the answer choices and see whether you can backsolve, eliminating obviously wrong answers.
(A) x = ¼ produces an expression in which you’re raising ¼ (a proper positive fraction) to a fractional power (taking a root), which will result in a number closer to 1, but certainly not 2.
(B) same problem as in (A).
(C) x = 1 produces 1^¼ = 1, not 2.
(D) x = 2 produces exactly what we want: 2^(22/4)=2^1 = 2.
We can stop there, but you can check 4 quickly: the expression becomes 44, which is not 2.
The correct answer is D.