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

 It is currently 19 Feb 2019, 19:18

### 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 February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT Prep Hour

February 20, 2019

February 20, 2019

08:00 PM EST

09:00 PM EST

Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST

February 21, 2019

February 21, 2019

10:00 PM PST

11:00 PM PST

Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

# The expression x[n]y is defined for positive values of x and

Author Message
TAGS:

### Hide Tags

Manager
Joined: 29 Nov 2011
Posts: 76
The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

09 May 2012, 03:43
5
17
00:00

Difficulty:

95% (hard)

Question Stats:

52% (03:14) correct 48% (03:04) wrong based on 360 sessions

### HideShow timer Statistics

The expression x[n]y is defined for positive values of x and y and for positive integer values of n as follows:

x[1]y = x^y
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y

If y = ½ and x[4]y = 2, then x =

A. ¼
B. ½
C. 1
D. 2
E. 4
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

09 May 2012, 05:50
3
6
Smita04 wrote:
The expression x[n]y is defined for positive values of x and y and for positive integer values of n as follows:
x[1]y = x^y
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y

If y = ½ and x[4]y = 2, then x =

A. ¼
B. ½
C. 1
D. 2
E. 4

Given:
$$x[1]y = x^y$$
If $$n$$ is even, $$x[n]y = (x[n-1]y)^x$$ (notice that it's the same as: if n is odd, x[n+1]y = (x[n]y)^x);
If $$n$$ is odd, $$x[n]y = (x[n-1]y)^y$$ (notice that it's the same as: if n is even, x[n+1]y = (x[n]y)^y)

Since 4 is even then $$x[4]y=(x[3]y)^x$$;
Since 3 is odd then $$(x[3]y)^x=((x[2]y)^y)^x=(x[2]y)^{xy}$$;
Since 2 is even then $$(x[2]y)^{xy}=((x[1]y)^x)^{xy}=(x[1]y)^{x^2y}$$;
Since $$x[1]y = x^y$$ then $$(x[1]y)^{x^2y}=(x^y)^{x^2y}=x^{x^2y^2}$$;

So, finally we have that: $$x[4]y=x^{x^2y^2}=2$$. Now, as $$y=\frac{1}{2}$$ then $$x^{\frac{x^2}{4}}=2$$ --> raise to fourth power: $$x^{x^2}=16$$ --> $$x=2$$ or $$x=-2$$.

_________________
##### General Discussion
SVP
Joined: 24 Jul 2011
Posts: 1530
GMAT 1: 780 Q51 V48
GRE 1: Q800 V740
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

09 May 2012, 04:51
x[4]y = (x[3]y)^y = ((x[2]y)^x)^y = ((((x[1]y)^y)^x)^y) = (x[1]y)^(x*y^2) = (x[1]y)^x

Now x[1]y = x^(1/2)

=> (x^(1/2))^x = 2
=> x^(x/2) = 2
=> x^x = 4
=> x = 2

Option D
_________________

GyanOne | Top MBA Rankings and MBA Admissions Blog

Premium MBA Essay Review|Best MBA Interview Preparation|Exclusive GMAT coaching

Get a FREE Detailed MBA Profile Evaluation | Call us now +91 98998 31738

Director
Status: Everyone is a leader. Just stop listening to others.
Joined: 22 Mar 2013
Posts: 769
Location: India
GPA: 3.51
WE: Information Technology (Computer Software)
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

11 Aug 2013, 03:50
42% right guesses , really a tough problem
_________________

Piyush K
-----------------------
Our greatest weakness lies in giving up. The most certain way to succeed is to try just one more time. ― Thomas A. Edison
Don't forget to press--> Kudos
My Articles: 1. WOULD: when to use? | 2. All GMATPrep RCs (New)
Tip: Before exam a week earlier don't forget to exhaust all gmatprep problems specially for "sentence correction".

Intern
Joined: 05 Jun 2014
Posts: 6
Location: India
GMAT 1: 610 Q44 V34
WE: Engineering (Computer Software)
The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

24 Jan 2015, 21:52
Bunuel, how can x be = -2 ?

$$(x)^(x)^(2)$$ ----> $$-2^(-2)^2$$ = -1/16 .. Am I right?
Manager
Joined: 27 Oct 2013
Posts: 208
Location: India
Concentration: General Management, Technology
GMAT Date: 03-02-2015
GPA: 3.88
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

24 Jan 2015, 22:08
Well, I would say a time consuming question...

I followed top-to-bottom approach for this problem...

starting from X[1]Y, I calculated the values till X[4]Y
then substituted the given parameters...

Option D is correct..

Time taken ~ 3 minutes(I am very susceptible to silly mistakes, so I have to be double sure)
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

25 Jan 2015, 02:43
saleem1992 wrote:
Bunuel, how can x be = -2 ?

$$(x)^(x)^(2)$$ ----> $$-2^(-2)^2$$ = -1/16 .. Am I right?

No.

(-2)^(-2)^2 = (-2)^4 = 16.

If exponentiation is indicated by stacked symbols, the rule is to work from the top down, thus:

$$a^{m^n}=a^{(m^n)}$$ and not $${(a^m)}^n$$, which on the other hand equals to $$a^{mn}$$.
_________________
Intern
Joined: 02 Feb 2016
Posts: 3
Location: India
Concentration: Technology, General Management
WE: Analyst (Computer Software)
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

23 May 2016, 00:43
Bunuel,
Could you please explain by using the information given rather then changing the odd even expressions.
I used the info given rather manipulating it.
I am not getting the result.
Manager
Joined: 06 Oct 2015
Posts: 91
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 01:35
Bunuel wrote:
Smita04 wrote:
The expression x[n]y is defined for positive values of x and y and for positive integer values of n as follows:
x[1]y = x^y
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y

If y = ½ and x[4]y = 2, then x =

A. ¼
B. ½
C. 1
D. 2
E. 4

Given:
$$x[1]y = x^y$$
If $$n$$ is even, $$x[n]y = (x[n-1]y)^x$$ (notice that it's the same as: if n is odd, x[n+1]y = (x[n]y)^x);
If $$n$$ is odd, $$x[n]y = (x[n-1]y)^y$$ (notice that it's the same as: if n is even, x[n+1]y = (x[n]y)^y)

Since 4 is even then $$x[4]y=(x[3]y)^x$$;
Since 3 is odd then $$(x[3]y)^x=((x[2]y)^y)^x=(x[2]y)^{xy}$$;
Since 2 is even then $$(x[2]y)^{xy}=((x[1]y)^x)^{xy}=(x[1]y)^{x^2y}$$;
Since $$x[1]y = x^y$$ then $$(x[1]y)^{x^2y}=(x^y)^{x^2y}=x^{x^2y^2}$$;

So, finally we have that: $$x[4]y=x^{x^2y^2}=2$$. Now, as $$y=\frac{1}{2}$$ then $$x^{\frac{x^2}{4}}=2$$ --> raise to fourth power: $$x^{x^2}=16$$ --> $$x=2$$ or $$x=-2$$.

Hi, Bunuel!
I am unable to understand the colored line. How does power shift from Y to X in the first line and X to Y in the second line?
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 01:49
NaeemHasan wrote:
Bunuel wrote:
Smita04 wrote:
The expression x[n]y is defined for positive values of x and y and for positive integer values of n as follows:
x[1]y = x^y
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y

If y = ½ and x[4]y = 2, then x =

A. ¼
B. ½
C. 1
D. 2
E. 4

Given:
$$x[1]y = x^y$$
If $$n$$ is even, $$x[n]y = (x[n-1]y)^x$$ (notice that it's the same as: if n is odd, x[n+1]y = (x[n]y)^x);
If $$n$$ is odd, $$x[n]y = (x[n-1]y)^y$$ (notice that it's the same as: if n is even, x[n+1]y = (x[n]y)^y)

Since 4 is even then $$x[4]y=(x[3]y)^x$$;
Since 3 is odd then $$(x[3]y)^x=((x[2]y)^y)^x=(x[2]y)^{xy}$$;
Since 2 is even then $$(x[2]y)^{xy}=((x[1]y)^x)^{xy}=(x[1]y)^{x^2y}$$;
Since $$x[1]y = x^y$$ then $$(x[1]y)^{x^2y}=(x^y)^{x^2y}=x^{x^2y^2}$$;

So, finally we have that: $$x[4]y=x^{x^2y^2}=2$$. Now, as $$y=\frac{1}{2}$$ then $$x^{\frac{x^2}{4}}=2$$ --> raise to fourth power: $$x^{x^2}=16$$ --> $$x=2$$ or $$x=-2$$.

Hi, Bunuel!
I am unable to understand the colored line. How does power shift from Y to X in the first line and X to Y in the second line?

This is given in the definition of the function: x[n]y is defined for positive values of x and y and for positive integer values of n as follows:
x[1]y = x^y
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y
_________________
Manager
Joined: 06 Oct 2015
Posts: 91
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 04:11
Bunuel,
But you have converted x[n+1]y = (x[n]y)^x to x[n]y = (x[n-1]y)^y when n is odd and same when n is even.
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 04:35
1
NaeemHasan wrote:
Bunuel,
But you have converted x[n+1]y = (x[n]y)^x to x[n]y = (x[n-1]y)^y when n is odd and same when n is even.

Where did I do that? Maybe you are confused by double exponents?

Since 4 is even then x[4]y=(x[3]y)^x;

Since 3 is odd then (x[3]y)^x=((x[2]y)^y)^x=(x[2]y)^(xy);

Since 2 is even then (x[2]y)^(xy)=((x[1]y)^x)^(xy)=(x[1]y)^(x^2y);
_________________
Manager
Joined: 10 Oct 2018
Posts: 128
Location: United States
GPA: 4
WE: Human Resources (Human Resources)
The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

18 Oct 2018, 00:00
Bunuel I've got doubt regarding the answer you provided.

Given:
$$x[1]y = x^y$$
If $$n$$ is even, $$x[n]y = (x[n-1]y)^x$$ (notice that it's the same as: if n is odd, x[n+1]y = (x[n]y)^x);
If $$n$$ is odd, $$x[n]y = (x[n-1]y)^y$$ (notice that it's the same as: if n is even, x[n+1]y = (x[n]y)^y)

how did you change the equation when the question stem states :
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y

is there any other easy way to solve this question as it seems to be really difficult! Please help. Thanks a lot in advance
_________________

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8883
Location: Pune, India
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

18 Oct 2018, 01:15
1
Smita04 wrote:
The expression x[n]y is defined for positive values of x and y and for positive integer values of n as follows:

x[1]y = x^y
If n is odd, x[n+1]y = (x[n]y)^x
If n is even, x[n+1]y = (x[n]y)^y

If y = ½ and x[4]y = 2, then x =

A. ¼
B. ½
C. 1
D. 2
E. 4

$$x[4]y = (x[3]y)^x = (x[2]y)^{xy} = (x[1]y)^{x^2y} = (x^y)^{x^2y} = x^{(xy)^2}$$

Since y = 1/2, we get $$2 = x^{(x/2)^2}$$

Look for a value of x from the options which will satisfy this equation. Since the power has x/2 but LHS is an integer, x should be a multiple of 2.

Try x = 2, you get $$2 = 2^{(2/2)^2} = 2^1$$
Satisfies

_________________

Karishma
Veritas Prep GMAT Instructor

Re: The expression x[n]y is defined for positive values of x and   [#permalink] 18 Oct 2018, 01:15
Display posts from previous: Sort by