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I'm confronted with the silliest questions in my mind regarding fractions raised to exponents. Can any one please confirm if my understanding is correct?
a) The result of a fraction between 0 to -1 raised to an exponent between 0 to 1 will be smaller Eg. (-1/2)^(1/2)
[I'm thinking the result will be an imaginary number but at the same time compelled to see this as -(1/2)^(1/2). Same for (c), (d) and (e)]
b) The result of a fraction between 0 to 1 raised to an exponent between 0 to -1 will be bigger
c) The result of a fraction between 0 to -1 raised to an exponent between 0 to -1 will be smaller
d) The result of a fraction between -1 to -2 raised to an exponent between 1 to 2 will be smaller
e) The result of a fraction between -1 to -2 raised to an exponent between -1 to -2 will be bigger
f) The result of a fraction between 1 to 2 raised to an exponent between -1 to -2 will be smaller
And the ones which are apparent...
g) The result of a fraction between 0 to 1 raised to an exponent between 0 to 1 will be bigger Eg. (1/2)^(1/2)
h) The result of a fraction between 1 to 2 raised to an exponent between 1 to 2 will be bigger
a) The result of a fraction between 0 to -1 raised to an exponent between 0 to 1 will be smaller Eg. (-1/2)^(1/2)
[I'm thinking the result will be an imaginary number but at the same time compelled to see this as -(1/2)^(1/2). Same for (c), (d) and (e)]
b) The result of a fraction between 0 to 1 raised to an exponent between 0 to -1 will be bigger
c) The result of a fraction between 0 to -1 raised to an exponent between 0 to -1 will be smaller
d) The result of a fraction between -1 to -2 raised to an exponent between 1 to 2 will be smaller
e) The result of a fraction between -1 to -2 raised to an exponent between -1 to -2 will be bigger
f) The result of a fraction between 1 to 2 raised to an exponent between -1 to -2 will be smaller
And the ones which are apparent...
g) The result of a fraction between 0 to 1 raised to an exponent between 0 to 1 will be bigger Eg. (1/2)^(1/2)
h) The result of a fraction between 1 to 2 raised to an exponent between 1 to 2 will be bigger
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19 Apr 2011, 21:31
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b c d e f are correct inferences
a is to be taken with a pinch of salt. the number can be imaginary rendering a useless conclusion.
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a is to be taken with a pinch of salt. the number can be imaginary rendering a useless conclusion.
Posted from my mobile device
20 Apr 2011, 12:38
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mniyer
These are certainly worthwhile questions to think about, since many GMAT inequality questions test your understanding of exponents. That said, some of your questions don't have an answer, because often the result will be undefined. In particular, raising negative numbers to fractional exponents often produces a result which is undefined (-2)^(1/2) is the square root of a negative number, for example, so doesn't make sense); you won't encounter this kind of situation on the GMAT.
I highlighted in red above something you will want to be careful about. If you see, for example, (-3)^2, that is not equal to -(3^2) - you can't take the negative sign out; in the same way, the simplification you've attempted above is not correct.
Of your questions above, the ones that are frequently important on the GMAT are b), f), g) and h). When your base is negative, then you will almost always only be concerned about understanding what will happen when your exponent is an integer. It then becomes important whether that exponent is even or odd.
If you have a negative base and some kind of fractional exponent - and offhand I don't know that I've seen an official question with that combination - that exponent will need an odd denominator, so if you want to think through what might happen in those cases, you should imagine what will happen if your exponent is 1/3 or -1/3.
