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Which of the following must be equal to zero for all real numbers x ?
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Bunuel
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Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  22 Jan 2014, 01:28
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Which of the following must be equal to zero for all real numbers x ?

I. \(-\frac{1}{x}\)

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and III only
(D) II and III only
(E) I, II , and III
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  22 Jan 2014, 01:28
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SOLUTION

Which of the following must be equal to zero for all real numbers x ?

I. \(-\frac{1}{x}\)

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and II I only
(D) II and III only
(E) I, II , and III

I. \(-\frac{1}{x}\neq{0}\) for any real number.

II. x + (-x) = x - x = 0.

III. x^0=1, for all nonzero numbers.

Answer: B.
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  22 Jan 2014, 04:54
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I go for B.

i. -1/x: if divided by any real number, the answer will never be zero. If 'x' =0; the answer is 'undefined'
ii. x+(-x): any real number will yield a result of zero when substituted for x in the expression
iii. x^0: this yields 1 as an answer if x is any real number.
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  22 Jan 2014, 20:06
I. -1/x will not be 0 for any real number; The value will be undefined if x is 0;
II. The value will be 0 for any real number. (ex.,) 2 + (-2) = 2-2 = 0;
III. x^0 = 1 for real numbers;

II only.

Ans is (B)
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  22 Jan 2014, 22:34
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Once we eliminate i, we realize from the answer choices that we can ignore ii and look at iii. Once iii is eliminated, we know ii is the answer.
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  22 Jan 2014, 23:29
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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

Which of the following must be equal to zero for all real numbers x ?

I. \(-\frac{1}{x}\)

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and II I only
(D) II and III only
(E) I, II , and III



Lets look at III first - Anything raise to 0 is 1, so x^0 = 1.

Now eliminate any answer choice that has III in options. Options C, D, E are eliminated.

I -1/x will never be equal to zero for any real number, value will be undefined if x = o

Hence option A is eliminated. We are left only with B, so B is the right answer.

II x + (-x) will be zero for any real number = x. for example x= 2.6. 2.6 - 2.6 = 0
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  23 Jan 2014, 01:09
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Bunuel wrote:

Which of the following must be equal to zero for all real numbers x ?

I. \(-\frac{1}{x}\)

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and II I only
(D) II and III only
(E) I, II , and III

Problem Solving
Question: 55
Category: Arithmetic Properties of numbers
Page: 69
Difficulty: 600




I would go with B

-1/x can be -1/2,-1/4,...
(x)+(-x) will be 0 for all real numbers
x^0 will either yield 1 or be indeterminate for 0^0
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  23 Jan 2014, 06:12
Answer is B

1/0 is undefined so option A is never 0
also 0^0 is undefined or 1 so option C is also never 0

so we are left with option B only
Answer B
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  25 Jan 2014, 11:26
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SOLUTION

Which of the following must be equal to zero for all real numbers x ?

I. \(-\frac{1}{x}\)

II. x + (-x)

III. x^0

(A) I only
(B) II only
(C) I and II I only
(D) II and III only
(E) I, II , and III

I. \(-\frac{1}{x}\neq{0}\) for any real number.

II. x + (-x) = x - x = 0.

III. x^0=1, for all nonzero numbers.

Answer: B.
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  04 Dec 2014, 02:51
Isn't zero a real number? And if it is, \(0^0\) is 0 isn't it? Or is it undefined in GMAT?
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  04 Dec 2014, 02:56
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Blackbox wrote:
Isn't zero a real number? And if it is, \(0^0\) is 0 isn't it? Or is it undefined in GMAT?


0 is a real number.

0^0, in some sources equals to 1 (not 0), some mathematicians say it's undefined. But you won't need this for the GMAT because the case of 0^0 is not tested on the GMAT.
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  04 Dec 2014, 04:26
Bunuel wrote:
0 is a real number.But you won't need this for the GMAT because the case of 0^0 is not tested on the GMAT


Thanks! But it is interesting you say that because if 0 is a real number then I am forced to use 0^0 for statement (3), in which case, this question isn't a GMAT type question. Oh well, I guess I will just chalk this one up to "fringe maths".
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  04 Dec 2014, 04:28
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Blackbox wrote:
Bunuel wrote:
0 is a real number.But you won't need this for the GMAT because the case of 0^0 is not tested on the GMAT


Thanks! But it is interesting you say that because if 0 is a real number then I am forced to use 0^0 for statement (3), in which case, this question isn't a GMAT type question. Oh well, I guess I will just chalk this one up to "fringe maths".


In any way 0^0 does NOT equal 0.
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Re: Which of the following must be equal to zero for all real numbers x ? [#permalink] New post  14 Jan 2017, 18:10
1, \(\frac{-1}{x}\) # 0
2, x + (-x) = x - x = always 0 for any number
3, x^0 = 1 but not 0
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