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General Question fractions and exponents...

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General Question fractions and exponents...

by thp510 » Mon Nov 08, 2010 8:16 pm
Is it safe to say that no matter what the exponent number is, when comparing two different numbers less than 1, the base (which is a fraction) is what really matters in terms of determining which is larger? For example....

(.2)^88 is less than (.9)^2

Since .2 is less than .9, no matter how many times .2 is multiplied, it's always going to be less than any power of .9? So if I compare the following two,

(.6)^23 versus (.7)^12, I automatically know that (.7)^12 is going to be bigger since it has a larger base regardless of the exponent. Is this correct? Are there any outliers or special case? I'm trying to grasp the concept of fractions when powered by any number.

Thanks!
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by zachthegnome » Mon Nov 08, 2010 9:03 pm
I think you're missing the important observation that everytime your multiply a fraction by another fraction it gets smaller. In both of your examples, it looked like you were raising the smaller fraction to a higher power meaning the smaller fraction was getting smaller more times than the larger fraction.

So it's not the base determining the rate of the shrinking number but the exponent.
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by Ian Stewart » Tue Nov 09, 2010 2:21 pm
thp510 wrote:Is it safe to say that no matter what the exponent number is, when comparing two different numbers less than 1, the base (which is a fraction) is what really matters in terms of determining which is larger? For example....

(.2)^88 is less than (.9)^2

Since .2 is less than .9, no matter how many times .2 is multiplied, it's always going to be less than any power of .9? So if I compare the following two,

(.6)^23 versus (.7)^12, I automatically know that (.7)^12 is going to be bigger since it has a larger base regardless of the exponent. Is this correct? Are there any outliers or special case? I'm trying to grasp the concept of fractions when powered by any number.

Thanks!
No, that's not true in general: both the base and the power are important. Take, for example:

(0.5)^3 = (1/2)^3 = 1/8 = 0.125

and

(0.4)^2 = (4/10)^2 = 16/100 = 0.16

Here, the second expression has the smaller base and still gives us a larger value after we evaluate the power.

It's important to notice that, if your base is between 0 and 1, the larger the power gets, the smaller your overall value gets, so (0.4) > (0.4)^2 > (0.4)^3 and so on.

It would not be an easy problem to compare two really awkward decimals with large powers. For example, it would not be straightforward (I'm just pulling numbers out of the air) to decide whether (0.35)^17 is larger than (0.28)^13; you just can't compare those easily, and you won't need to do that kind of thing on the GMAT, so it's not something you should be concerned about.

There is one case where the comparison is straightforward: if our bases are between 0 and 1, and our powers are positive, if you raise the smaller base to the larger power, that will give you the smallest value. This is the reason in both your examples ((0.2)^88 < (0.9)^2 and (0.6)^23 < (0.7)^12), you could make the comparison; the number with the smaller base also had the larger power. So for example, (0.15)^31 is definitely less than (0.19)^27, because (0.15)^31 < (0.15)^27 (the term with the smaller power is larger, as we saw above) and (0.15)^27 is less than (0.19)^27 (because the powers are the same and the bases are positive, we can just compare the bases).
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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