# Fun with GRE Exponents

##### February 21, 2014

Grab a piece of paper and try this exercise (I promise that it is relevant to an important lesson on GRE exponents.) Fold it in half, then in half again. And again. And again. You’ll probably only be able to fold it about seven or eight times before it simply becomes impossible to fold anymore. But you will notice that the wad of paper that you’re holding in your hand is significantly thicker than the thickness of the original piece of paper. Now imagine that you had a wider piece of paper that you could fold in half 50 times…how thick do you think this hypothetical wad of paper would be after folding it 50 times? Give it a guess — three inches thick? six inches thick? Even the most ambitious guess is usually still less than ten feet.

What’s the answer? The distance from the Earth to the Sun. Seems impossible, doesn’t it? Let’s explore the underlying math. The thickness of a typical piece of paper is usually around 1/200th of an inch. Every time you fold that piece of paper, you are doubling its thickness. So you have the original thickness of the paper — times 2, times 2, times 2, etc. So eventually you have multiplied the original thickness of the paper times 250 . That number — 250 — is so extraordinarily enormous, that even when multiplied by something as tiny as 1/200, gives you a number that, in inches, is a little bit more than the distance from the Earth to the Sun! So it was, in a way, a trick question, since there is no way a piece of paper can actually be folded this many times.

In fact, most calculators don’t even have enough digits to display the numerical value of 250 . So, even though you have an on-screen calculator on the GRE, it wouldn’t be helpful in calculating a value here. However, remember that the GRE Quantitative section is not a test on numbers — it is a test on concepts. So when you see base numbers raised to huge exponents on the GRE, don’t even think about actually sitting there and calculating the value! A previous blog entry showcased an example of how to deal with exponents efficiently in Quantitative Comparisons. Let’s now take a look at an example from Problem Solving:

If 426 = 16(x+1) , what is the value of x?

If you were a masochist (or, let’s face it, a pretty clumsy GRE test taker), you may actually consider calculating out 426 . This would undoubtedly not only take up an entire 35-minute Quantitative section of the GRE, but more likely the better part of an afternoon, so a Kaplan-trained student wouldn’t even consider it! To use any exponent rules, you must make sure that the base numbers are the same throughout the problem, and right now, that is not the case.

How do you turn 16(x+1) into something with a base number of 4? Is there a way you can express 16 as 4 raised to some power? Certainly: 16 = 42.

Thus, 16(x+1) can be re-written as (4)(x+1) .

Exponent rules tell you that when exponents are right next to each other, you multiply them, and this becomes 4(2x+2) , and you now have:

426 = 42x+2

Now, drop out the 4’s and voila — a much more pleasant looking algebra problem:

26 = 2x + 2,

which becomes 24 = 2x,

which becomes 12 = x.

So mastering exponent problems is not simply about memorizing a bunch of rules, though memorizing and understanding the rules is certainly a necessary component of your GRE preparation. Thinking critically about these exponent rules is what gives Kaplan students the ability to “raise their powers” on GRE Test Day!

Since 2002, Gene has been helping Kaplan GRE students become test-day champions. After graduating from the University of Maryland with a dual-degree in Political Science and Communications, Gene quickly set to work as one of Kaplan's best teachers. As an aficionado and master of all things test-prep, Gene's students agree that he not only helps demystify the GRE, but also makes studying for the test an enjoyable, and dare we say, fun experience as he coaches them towards GRE excellence. His expertise earned Gene an esteemed "Teacher of the Year" Award in 2006 and 2009 and has made him one of Kaplan's most requested GRE instructors. When he's not instilling Kaplan's proven test-day strategies to students around the world, Gene is an avid traveler, amateur magician, and guitar player. You can find Gene teaching GRE Classroom Anywhere or in the videos of GRE On Demand.

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