January 19, 2019 January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT. January 20, 2019 January 20, 2019 07:00 AM PST 07:00 AM PST Get personalized insights on how to achieve your Target Quant Score.
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 13 Jan 2018
Posts: 181
Location: India
GPA: 4

What is the value of x?
[#permalink]
Show Tags
03 Jan 2019, 19:34
Question Stats:
23% (00:33) correct 77% (00:50) wrong based on 64 sessions
HideShow timer Statistics
What is the value of x? 1) \(x^x = 1\) 2) \(x^a = 1\) where a > 0
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
____________________________ Regards,
Chaitanya +1 Kudos if you like my explanation!!!



Math Expert
Joined: 02 Aug 2009
Posts: 7200

Re: What is the value of x?
[#permalink]
Show Tags
04 Jan 2019, 08:31
eswarchethu135 wrote: What is the value of x?
1) \(x^x = 1\) 2) \(x^a = 1\) where a > 0 1) \(x^x = 1\) Let x be 0, \(0^0=1\), also 1^1=0 Thus x acb be 0 or 1 Insufficient 2) \(x^a = 1\) where a > 0 \(1^a=1\) and \((1)^a\), when a is even.. Thus x can be 1 or 1.. Insufficient Combined x is 1.. Sufficient C
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 616

Re: What is the value of x?
[#permalink]
Show Tags
04 Jan 2019, 11:42
eswarchethu135 wrote: What is the value of x?
1) \(x^x = 1\) 2) \(x^a = 1\) where a > 0
chetan2u ´s argument for the insufficiency of statement (2) is perfect. The SUFFICIENCY of statement (1) is proved below, although I must say the arguments are OUT of GMAT´s scope (in terms of abstraction, so to speak). Important: 0^0 is not equal to 1, it´s an undefined mathematical expression. (If you prefer, it does not exist, from the fact that it cannot be defined properly.) The reason people believe it is equal to 1 is related to the fact that in SOME cases it is convenient to use 0^0=1 as a CONVENTION. (The best example I know is when dealing with multiindexes, in the context of Partial Differential Equations, for instance.) All that put, let´s see the reason why (1) is SUFFICIENT and, therefore, the correct answer is (A): \({x^x} = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left {{x^x}} \right = 1\,\,\,\,\,\left( * \right)\) \(x > 0\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{ \,0 < x < 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} < 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = 1\,\,,\,\,\,0 < x < 1\,\,\,\,\,{\rm{impossible}}\,\, \hfill \cr \,x = 1\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{possible}} \hfill \cr \,x > 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} > \,\,\,1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = 1\,\,,\,\,\,x > 1\,\,\,\,\,{\rm{impossible}} \hfill \cr} \right.\) \(x < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = {1 \over {{x^{\left x \right}}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\mathop = \limits^{\left( * \right)} \,\,\left {{x^x}} \right\, = {1 \over {\,\left {{x^{\left x \right}}} \right}} = {1 \over {\left x \right{\,^{\left x \right}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left x \right{\,^{\left x \right}}\,\,\, = \,\,\,1\,\,\,\,\left\{ \matrix{ \,\,\,\left x \right \ne 1\,\,\,\,\mathop \Rightarrow \limits^{{\rm{previous}}\,\,{\rm{cases}}} \,\,\,\,\,{\rm{impossible}} \hfill \cr \,\,\,x =  1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x^x} =  1 \ne 1 \hfill \cr} \right.\) Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



Math Expert
Joined: 02 Aug 2009
Posts: 7200

Re: What is the value of x?
[#permalink]
Show Tags
04 Jan 2019, 19:01
fskilnik wrote: eswarchethu135 wrote: What is the value of x?
1) \(x^x = 1\) 2) \(x^a = 1\) where a > 0
chetan2u ´s argument for the insufficiency of statement (2) is perfect. The SUFFICIENCY of statement (1) is proved below, although I must say the arguments are OUT of GMAT´s scope (in terms of abstraction, so to speak). Important: 0^0 is not equal to 1, it´s an undefined mathematical expression. (If you prefer, it does not exist, from the fact that it cannot be defined properly.) The reason people believe it is equal to 1 is related to the fact that in SOME cases it is convenient to use 0^0=1 as a CONVENTION. (The best example I know is when dealing with multiindexes, in the context of Partial Differential Equations, for instance.) All that put, let´s see the reason why (1) is SUFFICIENT and, therefore, the correct answer is (A): \({x^x} = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left {{x^x}} \right = 1\,\,\,\,\,\left( * \right)\) \(x > 0\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{ \,0 < x < 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} < 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = 1\,\,,\,\,\,0 < x < 1\,\,\,\,\,{\rm{impossible}}\,\, \hfill \cr \,x = 1\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{possible}} \hfill \cr \,x > 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} > \,\,\,1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = 1\,\,,\,\,\,x > 1\,\,\,\,\,{\rm{impossible}} \hfill \cr} \right.\) \(x < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = {1 \over {{x^{\left x \right}}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\mathop = \limits^{\left( * \right)} \,\,\left {{x^x}} \right\, = {1 \over {\,\left {{x^{\left x \right}}} \right}} = {1 \over {\left x \right{\,^{\left x \right}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left x \right{\,^{\left x \right}}\,\,\, = \,\,\,1\,\,\,\,\left\{ \matrix{ \,\,\,\left x \right \ne 1\,\,\,\,\mathop \Rightarrow \limits^{{\rm{previous}}\,\,{\rm{cases}}} \,\,\,\,\,{\rm{impossible}} \hfill \cr \,\,\,x =  1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x^x} =  1 \ne 1 \hfill \cr} \right.\) Regards, Fabio. I would argue against someone claiming 0^0 is not 1. Yes, 0^0 can be called undefined at some places but 0^0 is taken as 1 in many important theorems and maths would break down if we did not accept this fact.. For example.. Binomial expansion.. \(7^2=(7+0)^2=7^20^0+7^10^1+7^00^2=49+0+0=49\) If we take 0^0 as undefined \(7^20^0\) should be undefined and not 49.. However 0^0 being debatable between 1 and undefined will not be tested on GMAT.
_________________
1) Absolute modulus : http://gmatclub.com/forum/absolutemodulusabetterunderstanding210849.html#p1622372 2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html 3) effects of arithmetic operations : https://gmatclub.com/forum/effectsofarithmeticoperationsonfractions269413.html
GMAT online Tutor



Manager
Joined: 13 Jan 2018
Posts: 181
Location: India
GPA: 4

Re: What is the value of x?
[#permalink]
Show Tags
04 Jan 2019, 21:54
chetan2u wrote: fskilnik wrote: eswarchethu135 wrote: What is the value of x?
1) \(x^x = 1\) 2) \(x^a = 1\) where a > 0
chetan2u ´s argument for the insufficiency of statement (2) is perfect. The SUFFICIENCY of statement (1) is proved below, although I must say the arguments are OUT of GMAT´s scope (in terms of abstraction, so to speak). Important: 0^0 is not equal to 1, it´s an undefined mathematical expression. (If you prefer, it does not exist, from the fact that it cannot be defined properly.) The reason people believe it is equal to 1 is related to the fact that in SOME cases it is convenient to use 0^0=1 as a CONVENTION. (The best example I know is when dealing with multiindexes, in the context of Partial Differential Equations, for instance.) All that put, let´s see the reason why (1) is SUFFICIENT and, therefore, the correct answer is (A): \({x^x} = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left {{x^x}} \right = 1\,\,\,\,\,\left( * \right)\) \(x > 0\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{ \,0 < x < 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} < 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = 1\,\,,\,\,\,0 < x < 1\,\,\,\,\,{\rm{impossible}}\,\, \hfill \cr \,x = 1\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{possible}} \hfill \cr \,x > 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} > \,\,\,1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = 1\,\,,\,\,\,x > 1\,\,\,\,\,{\rm{impossible}} \hfill \cr} \right.\) \(x < 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{x^x} = {1 \over {{x^{\left x \right}}}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\mathop = \limits^{\left( * \right)} \,\,\left {{x^x}} \right\, = {1 \over {\,\left {{x^{\left x \right}}} \right}} = {1 \over {\left x \right{\,^{\left x \right}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left x \right{\,^{\left x \right}}\,\,\, = \,\,\,1\,\,\,\,\left\{ \matrix{ \,\,\,\left x \right \ne 1\,\,\,\,\mathop \Rightarrow \limits^{{\rm{previous}}\,\,{\rm{cases}}} \,\,\,\,\,{\rm{impossible}} \hfill \cr \,\,\,x =  1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{x^x} =  1 \ne 1 \hfill \cr} \right.\) Regards, Fabio. I would argue against someone claiming 0^0 is not 1. Yes, 0^0 can be called undefined at some places but 0^0 is taken as 1 in many important theorems and maths would break down if we did not accept this fact.. For example.. Binomial expansion.. \(7^2=(7+0)^2=7^20^0+7^10^1+7^00^2=49+0+0=49\) If we take 0^0 as undefined \(7^20^0\) should be undefined and not 49.. However 0^0 being debatable between 1 and undefined will not be tested on GMAT. There is a debate still going on about \(0^0\) is equal to 1 or undefined. But in GMAT sense and also in daily math usage anything power 0 is 1. This applies to \(0^0\) as well. So to avoid confusion its better to accept \(0^0\) as 1, atleast in GMAT.
_________________
____________________________ Regards,
Chaitanya +1 Kudos if you like my explanation!!!



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 616

Re: What is the value of x?
[#permalink]
Show Tags
05 Jan 2019, 16:11
In the GMAT the test taker will not be in the position to have to choose between the two options (1 or undefined). I am aware that in many cases it is useful to consider 0^0 = 1, as I explained in my previous post. Cheetan´s example is a good one, by the way. On the other hand, one cannot define 0^0 as 1 (or as any other real number) without coming into math contradictions, specially in Mathematical Analysis (the area in which I am a Ph.D. candidate). In other words, accepting 1 is the possibility that "gets into trouble", not the other choice. (That´s why considering 0^0 undefined is the "conservative pointofview".) Thank you all for your interest in the theme, Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net




Re: What is the value of x? &nbs
[#permalink]
05 Jan 2019, 16:11






