GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Dec 2018, 02:31

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### Free GMAT Strategy Webinar

December 15, 2018

December 15, 2018

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.

# Is m^n a perfect square?

Author Message
TAGS:

### Hide Tags

Intern
Joined: 02 Feb 2018
Posts: 31
Is m^n a perfect square?  [#permalink]

### Show Tags

Updated on: 01 Dec 2018, 05:24
2
00:00

Difficulty:

75% (hard)

Question Stats:

25% (00:41) correct 75% (00:50) wrong based on 81 sessions

### HideShow timer Statistics

Is $$m^n$$ a perfect square?

(1) m is a perfect square

(2) n is a perfect square

Originally posted by barryseal on 29 Nov 2018, 04:01.
Last edited by barryseal on 01 Dec 2018, 05:24, edited 2 times in total.
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 544
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

Updated on: 30 Nov 2018, 05:24
barryseal wrote:
Is $$m^n$$ a perfect square?

(1) m is a perfect square

(2) n is a perfect square

$${m^n}\,\,\mathop = \limits^? \,\,{K^2}\,\,\,,\,\,\,K\,\,{\mathop{\rm int}}$$

$$\left( 1 \right)\,\,m = {J^2},\,\,J\mathop \ge \limits^{{\rm{WLOG}}} 0\,\,\,{\mathop{\rm int}} \,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {4,0.5} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,n = {W^2},\,\,W\mathop \ge \limits^{{\rm{WLOG}}} 0\,\,\,\,{\mathop{\rm int}} \,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,{m^n} = {\left( {{J^2}} \right)^{{W^2}}} = {\left( {{J^{{W^2}}}} \right)^2}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\left( {{J^{{W^2}}} = K} \right)\,\,\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

POST-MORTEM (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 :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Originally posted by fskilnik on 29 Nov 2018, 08:03.
Last edited by fskilnik on 30 Nov 2018, 05:24, edited 1 time in total.
CEO
Joined: 11 Sep 2015
Posts: 3237
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

29 Nov 2018, 16:38
2
Top Contributor
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, 04:36
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: 544
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

30 Nov 2018, 05:25
Hi, Brent!

You are right. I have corrected my solution accordingly.

Regards,
Fabio.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

CEO
Joined: 11 Sep 2015
Posts: 3237
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

Updated on: 30 Nov 2018, 09:13
Top Contributor
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, 08:45.
Last edited by GMATPrepNow on 30 Nov 2018, 09:13, edited 2 times in total.
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 544
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

Updated on: 30 Nov 2018, 17:00
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 :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Originally posted by fskilnik on 30 Nov 2018, 09:09.
Last edited by fskilnik on 30 Nov 2018, 17:00, 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, 10:00
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, 10:08
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: 544
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

30 Nov 2018, 10:55
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 !

We all get things wrong sometimes. The best proof is the posts above (mine included, of course)!

Regards,
Fabio.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Intern
Joined: 19 Aug 2018
Posts: 28
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

30 Nov 2018, 17:22
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: 544
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

30 Nov 2018, 18:07
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 !

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 :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Intern
Joined: 19 Aug 2018
Posts: 28
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

30 Nov 2018, 18:10
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 !

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: 544
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

30 Nov 2018, 18:23
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 :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

Intern
Joined: 24 Jun 2018
Posts: 6
Location: India
GMAT 1: 720 Q47 V38
GPA: 3.6
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

02 Dec 2018, 22:36
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: 31
Re: Is m^n a perfect square?  [#permalink]

### Show Tags

02 Dec 2018, 23:11
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? &nbs [#permalink] 02 Dec 2018, 23:11
Display posts from previous: Sort by