All the major test prep books for the SAT, ACT, and GRE -- published by companies like Kaplan, Princeton Review, Barron's, and Manhattan Test Prep -- are poorly written, conceptually deficient, and, worst of all, riddled with serious errors. Students can't be expected to learn from books that aren't even right! And I don't mean the books are riddled simply with typos, which unfortunately is also true, because they are so poorly edited; I mean they really are riddled with serious conceptual errors.

Here's a simple example from the very beginning -- the diagnostic test, of all things! -- of Princeton Review's "1,014 GRE Practice Questions." The problem is on page 24, and the answer key and explanation is on page 38. Not only is their answer wrong; what's worse, their *explanation* is wrong, too! I'll set off the problem by dashes (----) and then add more commentary after.

NOTE: The question is a classic GRE "quantitative comparison," so it's hard to represent in HTML. The goal is this: for the two quantities, A and B, below, you must select which of the following is true: (A) quantity A is larger than quantity B, (B) quantity B is larger than quantity A, (C) the two quantities are equal, or (D) the relationship between the two quantities cannot be determined.

----

x and y are positive numbers

Quantity A: sqrt(x) - sqrt(y)

Quantity B: sqrt[x-2*sqrt(x*y)+y]

Princeton Review's Explanation: "The answer is (C): the two quantities are equal. Quantity B contains a common quadratic pattern. Factor the right-hand side sqrt[x-2*sqrt(x*y)+y]=sqrt[(sqrt(x)-sqrt(y))^2]=sqrt(x)-sqrt(y). Both quantities are equal, so the answer is choice (C)."

----

Princeton is right in saying that

x-2*sqrt(x*y)+y=(sqrt(x)-sqrt(y))^2

but they are absolutely wrong in saying that

sqrt[(sqrt(x)-sqrt(y))^2]=sqrt(x)-sqrt(y)

The worst part about this error isn't even that it is serious, but rather that it is commonly tested! It's not just that the Princeton Review writers made a conceptual mistake: they made the very sort of mistake that standardized tests like the GRE and GMAT are *designed* to test! Do yourself a favor -- don't waste your money on test prep books written by people who don't even know how to avoid the most commonly tested concepts. Hire an expert tutor instead!

## Comments

I think you have a typo in UR explanation.

If I understand correctly

Quantity B: sqrt[x-2*sqrt(x*y)+y]

Should be. sqrt[x-2+sqrt(x*y)+y]

Please let me know. Otherwise PR is both confusing AND mistaken.

Hi Jerry,

Thanks for writing, but I didn't make a typo. Of course, this post isn't the easiest thing to read because of the notation, but I didn't make a mistake. And I'm not sure why you think the multiplication (2 times the square root of xy) should be changed to addition (2 plus the square root of xy).

Best wishes,

Matt