The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a 700 level GMAT question on our blog. Today, they have followed up with the answer:

An efficient way to attack this problem is to rephrase the expression in the stem. Since a sixth power is a square, we can look at the difference of sixth powers as the difference of squares, and factor:

x6y6 = (x3)2 – (y3)2 = (x3 + y3)(x3y3)

This new expression is not among the answer choices, but as a search strategy, we should focus on these factors.

Choice (A) contains (x3 + y3), so we should determine whether the rest of (A) works out to (x3y3). The terms (x2 + y2)(xy) look promising, since they give us +x3 and –y3, but we get cross-terms that don’t cancel: (x2 + y2)(xy) = x3x2y + xy2y3.

Choice (B) is close but not right. (x3y3)(x3y3) = (x3y3)2, which also gives us cross-terms that don’t cancel: (x3y3)2 = x6 – 2x3y3 + y6. (Moreover, the sign is wrong on the y6 term.)

Let’s skip to choice (E), since we see (x3 + y3). The remaining terms multiply out as follows:

(x2 + xy + y2)(xy) = x3 + x2y + xy2x2yxy2y3 = x3y3. This is what we were looking for. We can verify that the other answer choices do not multiply out to x6y6 exactly.

The correct answer is (E).

Of course, you can also plug simple numbers. Don’t pick 0 for either x or y, because too many terms will cancel. Also, don’t pick the same number for x and y, because then the result is 0 (and every answer choice gives you 0 as well). But if you pick 2 and 1, you can keep the quantities relatively small and still eliminate wrong answers.

26 – 16 = 64 – 1 = 63. Our target is 63.

(A) (8 + 1)(4 + 2)(2 – 1) = 48

(B) (8 – 1)(8 – 1) = 49

(C) (4 + 2)(4 + 2)(2 + 1)(2 – 1) = 108

(D) (16 + 1)(2 + 1)(2 – 1) = 51

(E) (8 + 1)(4 + 2 + 1)(2 – 1) = 63

Again, the correct answer is (E).

Incidentally, a “zenzicube” is a very old word for the sixth power of a number. An even funnier word exists for the eighth power of a number: zenzizenzizenzic. The “zenzi” part means “square,” so a “zenzicube” is the square of a cube, or (x3)2 = x6. Likewise, a “zenzizenzizenzic” is the square of a square of a square, or ((x2)2)2 = x8. Modern exponential notation has its advantages.

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