Breaking Down A GMATPrep Exponents Problem
This weeks problem emphasizes the importance of recognizing common ways to manipulate exponent expressions efficiently and accurately. I wont say any more until after youve tried this GMATPrep problem for yourself.
Set your timer for 2 minutes. and GO!
[pmath](sqrt{9+sqrt{80}}+sqrt{9-sqrt{80}})^2=[/pmath](A) 1
(B) [pmath]9-4sqrt{5}[/pmath]
(C) [pmath]18-4sqrt{5}[/pmath]
(D) 18
(E) 20
Thats a pretty nasty equation, isnt it? If we're not careful, we could end up with some totally ugly thing that doesnt look anything like any of the answers. On this one, take a few seconds to think about shortcuts before you dive into manipulation.
The first things to notice are the outer parentheses and the exponent of 2: theyre asking us to square something. The inner part looks pretty annoying, but it also looks symmetrical in a way. If we ignore all those square root signs for a second, weve got 9 + 80 and 9 80. If that were truly ALL we had, this problem would be a lot easier, wouldnt it? Instead we have a bunch of square root signs.
But heres the thing: that doesnt mean that we shouldnt use the same general idea to solve!
Why would it be easier if the expression was (9+80)(9-80)? (besides the fact that you can just do the actual math inside the parentheses)? It is because that looks like one of the famous special products: (a+b)(a-b). Hmm.
(a+b)(a-b) = [pmath]a^2[/pmath] [pmath]b^2[/pmath]
How does that help? We may not be 100% sure how this will help, but it does offer a different way to write the equation that we currently have. And clearly were not going to try to calculate the actual numbers without a calculator the square root of 80? I dont think so. So, lets try using the special products to write this in a different way and see if that helps simplify things.
But the next thing to notice is that we cant apply our special products knowledge with the 9 and 80 as they appear in this problem. This is because the two sets of terms actually have a plus sign between them, not multiplication! The special products require multiplication between the two sets of terms (thats annoying). But were on the special products track, so what other special products might we be able to use here?
What about the whole big expression? We could write the whole thing as [pmath](a+b)^2[/pmath] , where
[pmath]a=sqrt{9+sqrt{80}}[/pmath] and [pmath]b=sqrt{9-sqrt{80}}[/pmath].
How would that help?Well,
[pmath](a+b)^2=a^2+2ab+b^2[/pmath]
so lets try doing that:
[pmath]a=sqrt{9+sqrt{80}}[/pmath] so
[pmath]a^2=(sqrt{9+sqrt{80}})^2=9+sqrt{80}[/pmath]
Remember, when we want to square something that has a square root sign over it, we just remove the square root sign. By the same token, [pmath]b^2=(sqrt{9-sqrt{80}})^2=9-sqrt{80}[/pmath]. Thats nice because both of those terms are not too bad. But next we move onto the middle term:
[pmath]2ab=2(sqrt{9+sqrt{80}})(sqrt{9-sqrt{80}})[/pmath]
That ones kind of ugly but it leaves us with our new full set of terms:
[pmath]9+sqrt{80}+2(sqrt{9+sqrt{80}})(sqrt{9-sqrt{80}})+9-sqrt{80}[/pmath]
Ugh. Can we simplify? You should notice that the first [pmath]sqrt{80}[/pmath] and the last one cancel out!. And we can add the two 9s. Now weve got
[pmath]18+2(sqrt{9+sqrt{80}})(sqrt{9-sqrt{80}})[/pmath]
Hopefully you can see that the stuff in the parentheses looks more like that other special product that we first thought of, (a+b)(a-b), because now weve actually got multiplication between the two sets of parentheses.
Thats good. But how do we handle those square root signs? We need to go back to our exponent rules. When we multiply two things that are both under square root signs, we can re-write everything under one big square root sign:
[pmath]18+2sqrt{(9+sqrt{80})(9-sqrt{80})}[/pmath]
That looks way better. Now, lets do the math thats under that big square root sign using this special product:
(a+b)(a-b) = [pmath]a^2-b^2[/pmath]
a = 9 and b = [pmath]sqrt{80}[/pmath]
[pmath]9^2-(sqrt{80})^2[/pmath]
8180=1
That whole ridiculous thing equals 1 thats amazing! Plug it back in to our big equation:
18+2(1) = 20
And the correct answer is E.
All of a sudden the problem seems a lot easier, doesnt it? Hindsight is always nice. The key to recognizing what to do on this one is paying attention to our instincts at the beginning. At first, it looks like there must be something to do with the special products because of the symmetrical-ness of the initial form until we notice that theres an addition sign between the two sets of terms instead of multiplication.
That might lead us to think we were on the wrong track entirely, but we werent. We just needed to think about the other special products. They put that symmetrical-ness there to get people to think about special products, but they werent going to make it that easy, so they made the first instinct something that really wouldnt work out in the end. What answer would you get if you took that initial [pmath](sqrt{9+sqrt{80}}+sqrt{9-sqrt{80}})^2[/pmath] and tried to simplify as though there were a multiplication sign between the two sets of terms instead of that addition sign? Well, when we did that math above (at the point in the problem when we really did have a multiplication sign), we got 1 as the answer. Plug that in here, and we get [pmath]1^2=1[/pmath]. Check it out thats answer A.
And look at answer D (18). Does that number look familiar? It should; its a partial answer. Thats what youd get if you did the first part semi-correctly, setting up [pmath]a=sqrt{9+sqrt{80}}[/pmath] and [pmath]b=sqrt{9-sqrt{80}}[/pmath] (which simplified to 18) correctly but then incorrectly dropping (or forgetting about) the whole messy middle part of the calculation that eventually simplified to 2, so wed have calculated only the 18 part, leading us to wrong answer D.
What are the three special products, by the way?
[pmath](a+b)^2=a^2+2ab+b^2[/pmath]
[pmath](a-b)^2=a^2-2ab+b^2[/pmath]
[pmath](a+b)(a-b)=a^2-b^2[/pmath]
These should definitely be on your list of things to memorize, if they arent already. After memorizing those basic forms, also pay attention to more non-traditional forms that youll see on harder problems. How do you recognize and decode these even when they look a lot more complicated than the three generic examples above?
Key Takeaways for Complicated Exponent Problems:
(1) When anything even superficially resembles something that might be a special product, look again. Be flexible in your adaptation of these special forms. If there are parentheses and any common terms, you want to take advantage of the shortcuts offered by the special products!
(2) While studying, pay attention to the more unusual presentations of the three special products. Especially on harder questions, theyre going to try to disguise the usage of these. Higher-level test-takers wont be fooled they learn to recognize special products in any form.
(3) Be very careful with your manipulation. The numbers are messy-looking on purpose; its easy to make a mistake. Also, dont be too afraid of those messy numbers look at how nicely they can work out in the end! (And we can already have an idea that this will have to happen when we glance at the answers they arent all that messy.)
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.