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Sub 505 (Easy)|   Exponents|      
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SonyGmat
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 17 Dec 2011, 12:21
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C
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If \(x^a*x^b=1\) and \(x\neq\) -1, 1, then \(a+b=\)?

A. x
B. -1
C. 0
D. 1
E. It cannot be determined

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I believe that the correct answer should be different. If the correct answer is the one that the Kaplan book says it is, how can we make sure that x isn't 0? I believe the correct answer should be E.

Any inputs?
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C
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? Updated on: 17 Dec 2011, 18:06
Originally posted by bhavinp on 17 Dec 2011, 13:07.
Last edited by bhavinp on 17 Dec 2011, 18:06, edited 1 time in total.
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This is an interesting question. I'll work through it for those that would like to see the entire problem worked out.

x^a * x^b can be simplified into x^(a + b)
So now we have x^(a + b) = 1

Given that x is not 1 or -1, the only way for x^(a + b) to equal 1 is if (a + b) = 0. Because any non-zero number raised to 0 equals 1

Now, here's where it gets interesting...

First let's talk about the properties of 0. 0 raised to any non-zero number is 0. But what is 0 raised to 0? Well, there's dispute among mathematicians on this. Some will argue 0^0 = 0 and others will argue 0^0 = 1. Because of this ambiguity the GMAT does not test the concept of 0^0.

So back to our question. We know that x^(a + b) = 1.

Let's say you belong to the group that says 0^0 = 0. If this is the case, then you know that x cannot be 0. Because if x were 0, then no matter what (a + b) equals, it would be impossible to get 1, but the question clearly states that x^(a + b) = 1. So x could not be 0 and this would not need to be specified in the question.

Let's say you belong to the group that says 0^0 = 1. If this is the case, then you know that (a + b) = 0. Because for you, any number raised to 0 = 1.

So in either case, C (0) is still the correct answer. But, while I can't be sure, I don't think the GMAT would have a question such as this because of the ambiguity around 0^0.

We actually have a free video lesson available on exponents. Just jump ahead at 2:42 in the video to see the properties of base 0. https://gmat.magoosh.com/lessons/216-intro-to-exponents

I hope that helps!
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 17 Dec 2011, 16:44
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bhavinp

x^a + x^b can be simplified into x^(a + b)
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There's a typo here: x^a + x^b most certainly is not equal to x^(a+b). I'm sure you meant to write (x^a)(x^b), which is equal to x^(a+b).
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 17 Dec 2011, 17:28
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And I'd add that I find some prep books make far too big a deal about technicalities about the number zero, technicalities the GMAT never actually tests. If you go through, for example, every DS question in OG12 and the green Quant Review book, there isn't a single question where you'd get a different answer if you forgot to consider zero.

Of course the number 0 can show up in questions, just as any other number can, but you won't ever need to concern yourself with what 0^0 might mean, or what 0! (zero factorial) is equal to. The GMAT isn't interested in obscure technicalities like those. So if you see x^0 on the GMAT, you can safely assume that it is equal to 1. If you see n! on the GMAT, you can assume n is positive. If you see the phrase "x is an even integer", it will never matter if x might be 0 (zero is just like any other even integer where addition, subtraction or multiplication are concerned). And if you see the phrase "the integer x is divisible by 7", it will never matter if x might be 0 (zero is just like any other multiple of 7 as far as divisibility properties go).

The only property of zero I see tested frequently is the following: if you multiply numbers together, and the result is zero, you must have multiplied by zero. So if abc = 0, at least one of a, b or c must be zero. Or if (x-1)(x-2) = 0, then either x - 1 = 0 or x - 2 = 0, so x is either 1 or 2. Beyond that one (very important) property, I can't think of any situations on the GMAT where you'd need to consider zero in any special way. I have seen quite a few prep company questions where zero is used as a 'trap', but I find those questions very unrealistic for the most part.
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 17 Dec 2011, 18:07
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@IanStewart - You are absolutely right about my typo. I fixed it in the original post. Thanks for pointing it out!
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 17 Dec 2011, 18:30
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@bhavinp: I believe that 0^0 is undefined. Therefore, I am not sure whether we should assume that it equals to 0 or 1 or any other number.

@Ian: If we assume that the correct answer is indeed the official answer, C, then I still can't understand how can we ensure that x isn't 0. I understand that probably this won't be a GMAT question that someone would possibly face, but as far as mathematics is concerned, should't the answer be E?
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 17 Dec 2011, 18:48
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SonyGmat
@bhavinp: I believe that 0^0 is undefined. Therefore, I am not sure whether we should assume that it equals to 0 or 1 or any other number.

@Ian: If we assume that the correct answer is indeed the official answer, C, then I still can't understand how can we ensure that x isn't 0. I understand that probably this won't be a GMAT question that someone would possibly face, but as far as mathematics is concerned, should't the answer be E?
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First, you definitely will never need to worry about this issue on the GMAT, so it isn't really worth spending much time on, but in any case, logically it doesn't change the answer if you consider the possibility that x=0 here. We know that

x^(a + b) = 1

Can x be equal to 0? Certainly not. 0^(a+b) is always either equal to 0 (when a+b is positive), or is undefined (when a+b is 0 or less). So 0^(a+b) is never equal to 1, and the equation above just won't ever be true if x=0.
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 18 Dec 2011, 07:55
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@sonygmat, x cannot be 0, because if x were 0 then x^(a+b) could not equal 1. The only way the answer could be E is if there were some value other than 0 for (a+b) such that x^(a+b) equals 1.

The more important point as Ian mentioned is that the GMAT does not test this concept.
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 19 Dec 2011, 03:35
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(x^a)(x^b)=1
x^(a+b)=1
which is possible only when a+b=0

in case x=0 then it will be non determinate form while we have expression equals to 1/
ans is C
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If x^a*x^b = 1 and x ≠ -1, 1, then a + b = ? 08 Jul 2025, 09:44
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