Manhattan GMAT Challenge Problem of the Week – 15 January 2013

by on January 15th, 2013

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If x^3 = 25, y^4 = 64, and z^5 = 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


First, rewrite the right sides of the given equations as powers of small integers:
x^3 = 25 = 5^2
y^4 = 64 = 2^6
z^5 = 216 = 6^3 (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 = 5^(2/3)
y = 2^(6/4) = 2^(3/2)
z = 6^(3/5)

Now compare the results in pairs, putting “??” between to indicate the unknown inequality. First, compare x and y:
5^(2/3) ?? 2^(3/2)

Raise both sides to the 6th power, to remove the fractional powers:
5^(2/3)^6 ?? 2^(3/2)^6
5^4 ?? 2^9
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:
2^(3/2) ?? 6^(3/5)

Raise both sides to the 10th power, to remove the fractional powers:
2^(3/2)^10 ?? 6^(3/5)^10
2^15 ?? 6^6

2^15 ?? 2^6 × 3^6
2^9 ?? 3^6
512 ?? (3^2)^3 = 9^3 = 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 6^9 is bigger than 5^10, 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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