Memorize This and Pick Up 2 or 3 Quant Questions on the Test!
Memorize what? Im not going to tell you yet.
Try this problem from the GMATPrep free practice tests first and see whether you can spot the most efficient solution.
* If xy = 1, what is the value of [pmath]{2^{(x+y)^2}}/{2^{(x-y)^2}}[/pmath]?(A) 2
(B) 4
(C) 8
(D) 16
(E) 32
All right, have you got an answer? How satisfied are you with your solution? If you did get an answer but you dont feel as though you found an elegant solution, take some time to review the problem yourself before you keep reading.
Step 1: Glance Read Jot
Take a quick glance; what have you got? PS. A given equation, xy = 1. A seriously ugly-looking equation. Some fairly nice numbers in the answers. Hmm, maybe you should work backwards from the answers?
Jot the given info on the scrap paper.
Step 2: Reflect Organize
Oh, wait. Working backwards isnt going to workthe answers dont stand for just a simple variable.
Okay, whats plan B? Does anything else jump out from the question stem?
Hey, those ugly exponents there is one way in which theyre kind of nice.
Theyre both one of the three common special products. In general, when you see a special product, try rewriting the problem usually the other form of the special product.
Step 3: Work
Heres the original expression again:
[pmath]{2^{(x+y)^2}}/{2^{(x-y)^2}}[/pmath]
Lets see.
[pmath](x+y)^2=x^2+2xy+y^2[/pmath]
[pmath](x-y)^2=x^2-2xy+y^2[/pmath]
Interesting. I like that for two reasons. First of all, a couple of those terms incorporate xy and the question stem told me that xy = 1, so maybe Im heading in the right direction. Heres what Ive got now:
[pmath](x+y)^2=x^2+2+y^2[/pmath]
[pmath](x-y)^2=x^2-2+y^2[/pmath]
And that takes me to the second reason I like this: the two sets of exponents look awfully similar now, and they gave me a fraction to start. In general, were supposed to try to simplify fractions, and we do that by dividing stuff out.
[pmath]{2^{x^2+2+y^2}}/{2^{x^2-2+y^2}}[/pmath]
How else can I write this to try to divide the similar stuff out? Wait, Ive got it:
The numerator: [pmath](2^x^2)(2^2)(2^y^2)[/pmath]
The denominator:[pmath](2^x^2)(2^-2)(2^y^2)[/pmath]
Theyre almost identical! Both of the [pmath](2^x^2)[/pmath]terms cancel out, as do the [pmath](2^y^2)[/pmath]terms, leaving me with:
[pmath]{2^2}/{2^-2}[/pmath]
I like that a lot better than the crazy thing they started me with. Okay, how do I deal with this last step?
First, be really careful. Fractions + negative exponents = messy. In order to get rid of the negative exponent, take the reciprocal of the base:
[pmath]{2^2}/{(1/2)^2}[/pmath]
Next, dividing by [pmath]1/2[/pmath] is the same as multiplying by 2:
[pmath]2^2*2^2[/pmath]
That multiplies to 16, so the correct answer is (D).
Key Takeaways: Special Products
(1) Your math skills have to be solid. If you dont know how to manipulate exponents or how to simplify fractions, youre going to get this problem wrong. If you struggle to remember any of the rules, start building and drilling flash cards. If you know the rules but make careless mistakes as you work, start writing down every step and pausing to think about where youre going before you go there. Dont just run through everything without thinking!
(2) You need to memorize the special products and you also need to know when and how to use them. The test writers LOVE to use special products to create a seemingly impossible question with a very elegant solution.Whenever you spot any form of a special product, write the problem down using both the original form and the other form. If youre not sure which one will lead to the answer, try the other form first, the one they didnt give you; this is more likely to lead to the correct answer (though not always).
(3) You may not see your way to the end after just the first step. Thats okay. Look for clues that indicate that you may be on the right track, such as xy being part of the other form. If you take a few steps and come up with something totally crazy or ridiculously hard, go back to the beginning and try the other path. Often, though, youll find the problem simplifying itself as you get several steps in.
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.