## Manhattan GMAT Challenge Problem of the Week – 15 January 2013

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## Question

If = 25, = 64, and = 216, and

xy> 0, which of the following is true?A.

x>y>z

B.y>x>z

C.y>z>x

D.z>y>x

E.z>x>y

## Answer

First, rewrite the right sides of the given equations as powers of small integers:

= 25 =

= 64 =

= 216 = (you might not have recognized this one, but you could get there with prime factorization)

Take roots as necessary to isolate each variable. You know that *x* and *z* must be positive, because their odd powers are positive; moreover, since *xy* > 0, *y* must also be positive.

Third root:

*x* =

*y* = =

*z* =

Now compare the results in pairs, putting “??” between to indicate the unknown inequality. First, compare *x* and *y*:

??

Raise both sides to the 6th power, to remove the fractional powers:

??

??

625 ?? 512

Since 625 is bigger than 512, *x* is bigger than *y*. You can now eliminate B, C, and D.

The remaining two answers differ as to whether *z* is largest or smallest. Since *y* seems simpler than *x* (both in the base and in the exponent), compare *y* and *z*:

??

Raise both sides to the 10th power, to remove the fractional powers:

??

??

?? ×

??

512 ?? = = 81 × 9 = 729

So *z* is larger than *y*, eliminating A. The answer must therefore be E.

Incidentally, it’s very difficult to prove by hand that *z* is bigger than *x*. To do so, you need to show that is bigger than , very much a non-trivial task! Fortunately, you don’t need to determine whether *z* or *x* is larger in order to get the right answer.

**The correct answer is E.**

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