Fun with GRE Exponents
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 (42 )(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!