May 19 07:00 PM EDT  08:00 PM EDT Some of what you'll gain: Strategies and techniques for approaching featured GMAT topics. Sunday May 19th at 7 PM ET May 19 07:00 AM PDT  09:00 AM PDT Get personalized insights on how to achieve your Target Quant Score. Sunday, May 19th at 7 AM PT May 20 10:00 PM PDT  11:00 PM PDT Practice the one most important Quant section  Integer Properties, and rapidly improve your skills. May 24 10:00 PM PDT  11:00 PM PDT Join a FREE 1day workshop and learn how to ace the GMAT while keeping your fulltime job. Limited for the first 99 registrants. May 25 07:00 AM PDT  09:00 AM PDT Attend this webinar and master GMAT SC in 10 days by learning how meaning and logic can help you tackle 700+ level SC questions with ease.
Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 02 Feb 2018
Posts: 36

Is m^n a perfect square?
[#permalink]
Show Tags
Updated on: 01 Dec 2018, 06:24
Question Stats:
23% (01:13) correct 77% (01:10) wrong based on 86 sessions
HideShow timer Statistics
Is \(m^n\) a perfect square? (1) m is a perfect square (2) n is a perfect square
Official Answer and Stats are available only to registered users. Register/ Login.
Originally posted by barryseal on 29 Nov 2018, 05:01.
Last edited by barryseal on 01 Dec 2018, 06:24, edited 2 times in total.



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

Re: Is m^n a perfect square?
[#permalink]
Show Tags
Updated on: 30 Nov 2018, 06:24
barryseal wrote: Is \(m^n\) a perfect square?
(1) m is a perfect square
(2) n is a perfect square
\({m^n}\,\,\mathop = \limits^? \,\,{K^2}\,\,\,,\,\,\,K\,\,{\mathop{\rm int}}\) \(\left( 1 \right)\,\,m = {J^2},\,\,J\mathop \ge \limits^{{\rm{WLOG}}} 0\,\,\,{\mathop{\rm int}} \,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {4,0.5} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\) \(\left( 2 \right)\,\,n = {W^2},\,\,W\mathop \ge \limits^{{\rm{WLOG}}} 0\,\,\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\) \(\left( {1 + 2} \right)\,\,{m^n} = {\left( {{J^2}} \right)^{{W^2}}} = {\left( {{J^{{W^2}}}} \right)^2}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {{J^{{W^2}}} = K} \right)\,\,\,\) The correct answer is (C). This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. POSTMORTEM (important to the most careful students): \(0^0\) is an undefined expression in Math (and also in the GMAT, of course), hence the case \(m=0\) is implicitly associated with \(n \neq 0\). (The question stem implicitly implies that \(m^n\) exists!)
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net
Originally posted by fskilnik on 29 Nov 2018, 09:03.
Last edited by fskilnik on 30 Nov 2018, 06:24, edited 1 time in total.



CEO
Joined: 12 Sep 2015
Posts: 3716
Location: Canada

Re: Is m^n a perfect square?
[#permalink]
Show Tags
29 Nov 2018, 17:38
barryseal wrote: Is \(m^n\) a perfect square?
(1) m is a perfect square
(2) n is a perfect square Since there's no information to suggest that m and n are integers, the correct answer is not A. Statement 1: m is a perfect square Consider these conflicting cases that satisfy statement 1: CASE A) m = 9 and n = 2. In this case, m^n = 9^2 = 81. So, the answer to the target question is "YES m^n IS a perfect square." CASE B) m = 9 and n = 0.5. In this case, m^n = 9^0.5 = 3. So, the answer to the target question is "NO m^n is NOT a perfect square." Cheers, Brent
_________________
Test confidently with gmatprepnow.com



Manager
Joined: 26 Feb 2017
Posts: 89
Location: India
GPA: 3.99

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 05:36
GMATPrepNow wrote: barryseal wrote: Is \(m^n\) a perfect square?
(1) m is a perfect square
(2) n is a perfect square Since there's no information to suggest that m and n are integers, the correct answer is not A. Statement 1: m is a perfect square Consider these conflicting cases that satisfy statement 1: CASE A) m = 9 and n = 2. In this case, m^n = 9^2 = 81. So, the answer to the target question is "YES m^n IS a perfect square." CASE B) m = 9 and n = 0.5. In this case, m^n = 9^0.5 = 3. So, the answer to the target question is "NO m^n is NOT a perfect square." Cheers, Brent Hi Brent, The correct answer is C here. If we know both m and n are perfect squares, then m^n is not a perfect square. m = 25 n = 4 m^n is not a perfect square. Is this correct? Posted from my mobile device



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 06:25
Hi, Brent! You are right. I have corrected my solution accordingly. Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



CEO
Joined: 12 Sep 2015
Posts: 3716
Location: Canada

Re: Is m^n a perfect square?
[#permalink]
Show Tags
Updated on: 30 Nov 2018, 10:13
dollytaneja51 wrote: Hi Brent,
The correct answer is C here. If we know both m and n are perfect squares, then m^n is not a perfect square. m = 25 n = 4 m^n is not a perfect square.
Is this correct?
Posted from my mobile device EDIT: I first read the above as m^n IS a perfect square. See Fabio's response below. Cheers, Brent
_________________
Test confidently with gmatprepnow.com
Originally posted by GMATPrepNow on 30 Nov 2018, 09:45.
Last edited by GMATPrepNow on 30 Nov 2018, 10:13, edited 2 times in total.



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

Re: Is m^n a perfect square?
[#permalink]
Show Tags
Updated on: 30 Nov 2018, 18:00
dollytaneja51 wrote: If we know both m and n are perfect squares, then m^n is not a perfect square. m = 25 n = 4 m^n is not a perfect square.
Is this correct?
This is NOT correct. First of all: when both m and n are perfect squares, we are sure both are nonnegative integers. (Each one is the square of an integer, therefore a nonnegative integer.) From that, when a perfect square (m) is put to the power of a nonnegative ineteger (n), we may conclude that \(m^n\) is also a perfect square: \({m^n} = {\left( {{J^2}} \right)^n} = {\left( {{J^n}} \right)^2} = {\text{in}}{{\text{t}}^2}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {J,n \geqslant 0\,\,{\text{ints}}\,,\,{\text{not}}\,\,{\text{both}}\,\,{\text{zero}}} \right)\) In the example provided, please note that: \({25^4} = {\left( {{{25}^2}} \right)^2}\,\,\underline {{\rm{is}}} \,\,{\rm{a}}\,\,{\rm{perfect}}\,\,{\rm{square}}\) If you prefer (equivalently), \(\sqrt {{{25}^4}} = {25^2} = {\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,{25^4}\,\,\underline {{\rm{is}}} \,\,{\rm{a}}\,\,{\rm{perfect}}\,\,{\rm{square}}\) Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net
Originally posted by fskilnik on 30 Nov 2018, 10:09.
Last edited by fskilnik on 30 Nov 2018, 18:00, edited 2 times in total.



Manager
Joined: 26 Feb 2017
Posts: 89
Location: India
GPA: 3.99

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 11:00
Dear Moderator,
Looks like OA is incorrect for this question.
Posted from my mobile device



Manager
Joined: 26 Feb 2017
Posts: 89
Location: India
GPA: 3.99

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 11:08
fskilnik wrote: dollytaneja51 wrote: If we know both m and n are perfect squares, then m^n is not a perfect square. m = 25 n = 4 m^n is not a perfect square.
Is this correct?
This is NOT correct. First of all: when both m and n are perfect squares, we are sure both are integers. (Each one is the square of an integer, therefore an integer.) From that, when a perfect square (m) is to the power of an INTEGER (n), we may conclude that \(m^n\) is also a perfect square: \({m^n} = {\left( {{J^2}} \right)^n} = {\left( {{J^n}} \right)^2} = {{\mathop{\rm int}} ^2}\,\,\,\,\,\left( {J \ge 0\,\,{\mathop{\rm int}} ,\,\,J\,\,{\rm{and}}\,\,{\rm{n}}\,\,{\rm{not}}\,\,{\rm{both}}\,\,{\rm{zero}}} \right)\) In the example provided, please note that: \({25^4} = {\left( {{{25}^2}} \right)^2}\,\,\underline {{\rm{is}}} \,\,{\rm{a}}\,\,{\rm{perfect}}\,\,{\rm{square}}\) If you prefer (equivalently), \(\sqrt {{{25}^4}} = {25^2} = {\mathop{\rm int}} \,\,\,\,\, \Rightarrow \,\,\,\,\,{25^4}\,\,\underline {{\rm{is}}} \,\,{\rm{a}}\,\,{\rm{perfect}}\,\,{\rm{square}}\) Regards, Fabio. Thanks Fabio for the response. It will be a perfect square. I dont know why I was thinking perfect square will have only three factors. Posted from my mobile device



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 11:55
dollytaneja51 wrote: Thanks Fabio for the response. It will be a perfect square.
I dont know why I was thinking perfect square will have only three factors.
Hi dollytaneja51 ! Thanks for your reply. We all get things wrong sometimes. The best proof is the posts above (mine included, of course)! Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



Intern
Joined: 19 Aug 2018
Posts: 32

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 18:22
What is the correct answer of this question? Is it C or E? In this cae what happened if 0^0?, because 0 is a perfect square.
Thank you.



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 19:07
jorgetomas9 wrote: What is the correct answer of this question? Is it C or E? In this cae what happened if 0^0?, because 0 is a perfect square.
Thank you. Hi jorgetomas9 ! The correct answer is (C). I explained in my first post that the case 0^0 must be implicitly considered excluded. Reason: we were asked whether m^n is a perfect square, hence we may (and should) assume that m^n exists (=is defined). Regards, Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



Intern
Joined: 19 Aug 2018
Posts: 32

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 19:10
fskilnik wrote: jorgetomas9 wrote: What is the correct answer of this question? Is it C or E? In this cae what happened if 0^0?, because 0 is a perfect square.
Thank you. Hi jorgetomas9 ! The correct answer is (C). I explained in my first post that the case 0^0 must be implicitly considered excluded. Reason: we were asked whether m^n is a perfect square, hence we may (and should) assume that m^n exists (=is defined). Regards, Fabio. Thank you, very much I marked C too, only I had a doubt related with 0^0



GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935

Re: Is m^n a perfect square?
[#permalink]
Show Tags
30 Nov 2018, 19:23
jorgetomas9 wrote: Thank you, very much I marked C too, only I had a doubt related with 0^0
I am glad things are clear now. See you in other posts and success in your studies! Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our highlevel "quant" preparation starts here: https://gmath.net



Intern
Joined: 24 Jun 2018
Posts: 6
Location: India
GPA: 3.6

Re: Is m^n a perfect square?
[#permalink]
Show Tags
02 Dec 2018, 23:36
Need help. How can C be the right answer?
Say m = 9 (perfect square) n = 9 (perfect square)
m^n = 9^9 = 9*9*..... 9 times.. ths is not a perfect square.



Intern
Joined: 02 Feb 2018
Posts: 36

Re: Is m^n a perfect square?
[#permalink]
Show Tags
03 Dec 2018, 00:11
agpamit wrote: Need help. How can C be the right answer?
Say m = 9 (perfect square) n = 9 (perfect square)
m^n = 9^9 = 9*9*..... 9 times.. ths is not a perfect square. 9 is not a prime factor > 9^9 = 3^18 = perfect square Posted from my mobile device




Re: Is m^n a perfect square?
[#permalink]
03 Dec 2018, 00:11






