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Exponential Equation Problem

Expert replies
by BlindVision » Fri Mar 06, 2009 9:51 pm
If u is a positive integer, which of the following could be a negative number?

A) u^7-u^6
B) u^3+u^4+u^5
C) u^-9
D) u^-13+u^13
E) u^3-u^8

OA=E

I would like to believe that I have a good understanding of exponential properties but for some reason, I'm puzzled at how the answer is derived. I reasoned that odd exponents will result in a positive number, and even exponents can result in either a positive/negative number. So if the variable u is positive to begin with, then how come E is the answer and not A? I would appreciate any help. Thank you...
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Source: — Problem Solving |

Re: Exponential Equation Problem

by Ian Stewart » Fri Mar 06, 2009 10:02 pm
BlindVision wrote:If u is a positive integer, which of the following could be a negative number?

A) u^7-u^6
B) u^3+u^4+u^5
C) u^-9
D) u^-13+u^13
E) u^3-u^8

OA=E

I would like to believe that I have a good understanding of exponential properties but for some reason, I'm puzzled at how the answer is derived. I reasoned that odd exponents will result in a positive number, and even exponents can result in either a positive/negative number. So if the variable u is positive to begin with, then how come E is the answer and not A? I would appreciate any help. Thank you...
If u is a positive integer (or any positive number - it doesn't need to be an integer), then u^x will be positive no matter what x is. It's only if u is negative that there is a possibility for u^x to be negative. If u is negative, and x is an integer, then u^x will be negative if x is odd, and positive if x is even.

Now if u > 1, it will always be true that u^x > u^y if x > y. So if u > 1, u^8 > u^3, and u^3 - u^8 < 0.
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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Re: Exponential Equation Problem

by BlindVision » Fri Mar 06, 2009 10:20 pm
Ian Stewart wrote:If u is negative, and x is an integer, then u^x will be negative if x is odd, and positive if x is even.
Thanks Ian. Now I have a question about the above quote. Can you please explain a little further of why that happens? Maybe some more examples will make me understand this better. Your help is appreciated.
Last edited by BlindVision on Fri Mar 06, 2009 10:25 pm, edited 1 time in total.
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by cramya » Fri Mar 06, 2009 10:25 pm
Excellent explanation Ian!

If u is negative, and x is an integer, then u^x will be negative if x is odd, and positive if x is even

Let u = -2

(-2)^ odd exponent will always be negative

i.e (-2)^3= - 8 ; (-2)^5 = -32 etc....

(-2)^ even exponent will always be positive

i.e (-2)^2= 4 ; (-2)^4 = 16 etc....



Hope this helps!

Regards,
CR
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by BlindVision » Fri Mar 06, 2009 10:27 pm
You guys are too awesome! I got it now! Thanks Cramya!
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by Ian Stewart » Fri Mar 06, 2009 10:53 pm
I'd just add two things:

-at least for this question, you could plug in u = 2 to see that E can be negative. Of course, on a different question, with more awkward exponents, that might not be practical;

-I've always found algebraic rules easier to remember when I understand why they're true, and a great way to begin to understand algebraic rules is to see what happens when you plug in simple numbers. That's what cramya did above, and it's a quick and useful thing to do when you're learning, say, any of the exponent rules.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com
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