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

It is currently 15 Nov 2019, 02:42

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?  [#permalink]

Show Tags

New post 11 Mar 2019, 06:33
00:00
A
B
C
D
E

Difficulty:

  75% (hard)

Question Stats:

29% (01:11) correct 71% (01:26) wrong based on 38 sessions

HideShow timer Statistics

GMATH practice exercise (Quant Class 12)

If \(x>0\), what is the sum of the roots of the equation \(\,{x^{\sqrt x }} = {x^2}\) ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Senior Manager
Senior Manager
avatar
G
Joined: 25 Feb 2019
Posts: 336
Reviews Badge
Re: If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?  [#permalink]

Show Tags

New post 11 Mar 2019, 07:29
IMO E ,

after compare , we get √x= 2

means x = 4 (as x>0)
and also by trial , we see 1 satisfy the equation.

so sum = 5

Posted from my mobile device
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
Re: If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?  [#permalink]

Show Tags

New post 11 Mar 2019, 15:13
fskilnik wrote:
GMATH practice exercise (Quant Class 12)

If \(x>0\), what is the sum of the roots of the equation \(\,{x^{\sqrt x }} = {x^2}\) ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

\(x > 0\,\,,\,\,\,\,{x^{\sqrt x }} = {x^2}\,\,\,\left( * \right)\)

\(?\,\,:\,\,{\rm{sum}}\,\,{\rm{of}}\,\,{\rm{roots}}\)


\(x = 1\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{inspection}}} \,\,\,\,\left\{ \matrix{
\,\,{x^{\sqrt x }} = {1^{\sqrt 1 }} = 1 \hfill \cr
\,\,{x^2} = {1^2} = 1 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{first}}\,\,{\rm{root}}\)


\(0 < x < 1\,\,\,{\rm{or}}\,\,\,x > 1\,\,\,:\,\,\,\,\,\)

\(\left( * \right)\,\,\,\, \Rightarrow \,\,\,{x^{\sqrt x \, - \,2}} = 1 = {x^0}\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{base}}\,\, \ne \,\,0,1, - 1} \,\,\,\,\,\sqrt x - 2 = 0\,\,\,\, \Rightarrow \,\,\,\,\,x = 4\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{second}}\,\,{\rm{root}}\)


\(? = 1 + 4\)


The correct answer is (E).

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Senior Manager
Senior Manager
User avatar
S
Joined: 12 Sep 2017
Posts: 306
If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?  [#permalink]

Show Tags

New post Updated on: 12 Mar 2019, 17:38
Hello fskilnik!

Why did you start with "inspection" 1?

What I did was:

\(\,{x^{\sqrt x }} = {x^2}\)

\(({\sqrt x } = 2)^2\)

\(x = 4\)

Am I totally wrong?

Regards!

Thank you fskilnik !

Originally posted by jfranciscocuencag on 11 Mar 2019, 21:00.
Last edited by jfranciscocuencag on 12 Mar 2019, 17:38, edited 1 time in total.
Veritas Prep GMAT Instructor
User avatar
V
Joined: 16 Oct 2010
Posts: 9783
Location: Pune, India
Re: If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?  [#permalink]

Show Tags

New post 11 Mar 2019, 23:02
fskilnik wrote:
GMATH practice exercise (Quant Class 12)

If \(x>0\), what is the sum of the roots of the equation \(\,{x^{\sqrt x }} = {x^2}\) ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


When the base is 0 or 1 or -1, it is possible that the two terms are equal.
But x cannot be 0 here because 0^0 is not defined.
If x = 1. the two terms are equal.
x cannot be -1 here since x > 0

Another way the two terms will be equal is if the exponents are equal.
\(\sqrt{x} = 2\)
\(x = 4\)

Sum of the roots = 1 + 4 = 5

Answer (E)
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 935
If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?  [#permalink]

Show Tags

New post 12 Mar 2019, 04:14
1
jfranciscocuencag wrote:
Hello fskilnik!

Why did you start with "inspection" 1?

What I did was:

\(\,{x^{\sqrt x }} = {x^2}\)

\(({\sqrt x } = 2)^2\)

\(x = 4\)

Am I totally wrong?

Regards!

Hi jfranciscocuencag !

Let me add other (I believe) nice details to complement Karishma´s important contribution.

(I am glad you have joined us, VeritasKarishma! I hope everything is well with you and your family!)

Given a fixed constant (say) \(b\), we have learned at school that \(b^x = b^y\) implies \(x=y\), although this is not, in general, true.

Examples: (i) \(0^1 = 0^2\) but \(1=2\) is false. (ii) \((-1)^0 = (-1)^2\) but \(0=2\) is false...

The true statement is the following: \(b^x = b^y\) implies \(x=y\) when the base \(b\) is different from -1, 0, and 1 (as Karishma properly mentioned).

This is a consequence of the injectivity of the exponential function when the base \(b\) is positive and different from 1, meaning: different powers will give different values when the base is the same but different from -1,0, and 1.

That´s exactly the reason I had to check the case of the base equals to 1 separately, and I did by inspection, of course.

Please note that Karishma and I were able to "impose equal powers" when we were in the blue case above...

I hope things are clear now!

Regards and success in your studies,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMAT Club Bot
If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?   [#permalink] 12 Mar 2019, 04:14
Display posts from previous: Sort by

If x>0, what is the sum of the roots of the equation x^sqrt(x)=x^2?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne