Manhattan GMAT Challenge Problem of the Week – 14 May 2012
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Question
The expression x[n]y is defined for positive values of x and y and for positive integer values of n as follows:
x[1]y =
If n is odd, x[n+1]y =
If n is even, x[n+1]y =If y = ½ and x[4]y = 2, then x =
A. ¼
B. ½
C. 1
D. 2
E. 4
![(x[n]y)^x](http://www.beatthegmat.com/mba/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_978.5_750ad616f66d5ae61bb0a4c022e9b311.png "(x[n]y)^x")
![(x[n]y)^y](http://www.beatthegmat.com/mba/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_978.5_90b0389990a9a4bd68aa34ba65b88491.png "(x[n]y)^y")
Answer
Construct x[4]y in stages, working up from n = 1:
x[1]y = 
To figure out x[2]y, apply the second part of the definition, using n = 1 (odd):
x[2]y = ![(x[1]y^x=(x^y)^x=x^{yx}](http://www.beatthegmat.com/mba/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_1002_a2833c0ffc6c7fc24a2fbdde7f62d34b.png "(x[1]y^x=(x^y)^x=x^{yx}")
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):
x[3]y = ![(x[2]y)^x=(x^{yx})^x=x^{yxx}](http://www.beatthegmat.com/mba/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_978.5_dfcef9090f0617f080829bbd07054a3c.png "(x[2]y)^x=(x^{yx})^x=x^{yxx}")
(We’ll avoid even tinier exponents by simply writing xx for 
.)
Finally, the x[4]y expression will add another y to the exponent:
x[4]y = ![(x[3]y)^y=(x^{yxx})^y=x^{yyxx}](http://www.beatthegmat.com/mba/wp-content/plugins/wpmathpub/phpmathpublisher/img/math_978.5_0dd98dceb69be048fc88f22d09af6eae.png "(x[3]y)^y=(x^{yxx})^y=x^{yyxx}")
Now, we are told that y is ½ and that this crazy expression 
equals 2. Substituting in y = ½, you get 
.
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 (i.e., taking a root). This will result in a number closer to 1 than ¼ is, but it can’t be larger than 1, and it’s 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: 
. This is the answer.
We can stop there, but we can check 4 quickly: the expression becomes 
, which is not 2.
The correct answer is D.
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