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Without further ado, today’s sample problem comes from our friends at Manhattan GMAT:
Problem:
x6 – y6 =
(A) (x3 + y3)(x2 + y2)(x – y)
(B) (x3 – y3)(x3 – y3)
(C) (x2 + y2)(x2 + y2)(x + y)(x – y)
(D) (x4 + y4)(x + y)(x – y)
(E) (x3 + y3)(x2 + xy + y2)(x – y)
Solution:
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:
x6 – y6 = (x3)2 – (y3)2 = (x3 + y3)(x3 – y3)
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 (x3 – y3). The terms (x2 + y2)(x – y) look promising, since they give us +x3 and –y3, but we get cross-terms that don’t cancel: (x2 + y2)(x – y) = x3 – x2y + xy2 – y3.
Choice (B) is close but not right. (x3 – y3)(x3 – y3) = (x3 – y3)2, which also gives us cross-terms that don’t cancel: (x3 – y3)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)(x – y) = x3 + x2y + xy2 – x2y – xy2 – y3 = x3 – y3. This is what we were looking for. We can verify that the other answer choices do not multiply out to x6 – y6 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.
