Today’s GMAT challenge question comes from our friends at ManhattanGMAT. To help you with your GMAT studying, try to solve the problem on your own, and then read on for the explanation of its solution:
Problem
x8 – y8 =
(A) (x4 – y4)2
(B) (x4 + y4)(x2 + y2)(x + y)(x – y)
(C) (x6 + y2)(x2 – y6)
(D) (x4 – y4)(x2 – y2)(x – y)(x + y)
(E) (x2 – y2)4
Solution
You need to recognize the expression as a difference of squares. Like all other even powers, x8 is a square, equal to (x4)2, so we use the template a2 – b2 = (a + b)(a – b), with a = x4 and b = y4:
x8 – y8 = (x4)2 – (y4)2 = (x4 + y4)(x4 – y4)
We continue breaking down the second part of the resulting expression, which is also a difference of squares.
x8 – y8 = (x4)2 – (y4)2 = (x4 + y4)(x2 + y2)(x2 – y2)
And we’re not done yet, because the last expression is of course also a difference of squares!
x8 – y8 = (x4)2 – (y4)2 = (x4 + y4)(x2 + y2)(x2 – y2) = (x4 + y4)(x2 + y2)(x + y)(x – y)
This final product matches the expression in choice (B), so the correct answer is (B).
Plugging numbers is probably too time-consuming in this case. Among positive integers, only 0 and 1 are easy to compute the eighth power of (unless you’ve memorized that 28 = 256). Moreover, several of the answer choices are designed to give you 0 if you choose x = y = 1.
If you did plug in x = 2 and y = 1, then you would get the following for choice (B):
256 – 1 = 255 = (16 + 1)(4 + 1)(2 + 1)(2 – 1) = (17)(5)(3).
If you happen to know already that 28 = 256, then you could get 255 as your target number relatively quickly. Since 255 ends in 5, it must be divisible by 5. No choice besides (B) gives you 5 as a factor if you plug in x = 2 and y = 1, so you wouldn’t need to compute the actual values of every choice. However, it’s still the case that the best way to do this problem is to recognize the original expression as a difference of squares, and then factor.
Again, the correct answer is (B).
If you’ve been following our Challenge Problems, you might remember “Difference of Zenzicubes” from last October. At the end of the explanation, we mentioned that a “zenzizenzizenzic” is the square of a square of a square, or ((x2)2)2 = x8. This name was coined by Robert Recorde, who also introduced something far more useful: the equals sign (=).
http://en.wikipedia.org/wiki/Zenzizenzizenzic
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Read the full article: GMAT Tip: Zenzizenzizenzic Zurprise