GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 23 Jan 2020, 07:04

### 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

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

Author Message
TAGS:

### Hide Tags

Manager
Joined: 29 Nov 2011
Posts: 70
The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

09 May 2012, 04:43
6
24
00:00

Difficulty:

95% (hard)

Question Stats:

52% (03:12) correct 48% (03:00) wrong based on 292 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: 60627
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

09 May 2012, 06:50
5
7
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: 1917
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, 05:51
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
_________________

Awesome Work | Honest Advise | Outstanding Results

Reach Out, Lets chat!
Email: info at gyanone dot com | +91 98998 31738 | Skype: gyanone.services
Director
Status: Everyone is a leader. Just stop listening to others.
Joined: 22 Mar 2013
Posts: 694
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, 04:50
42% right guesses , really a tough problem
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, 22:52
Bunuel, how can x be = -2 ?

$$(x)^(x)^(2)$$ ----> $$-2^(-2)^2$$ = -1/16 .. Am I right?
Manager
Joined: 27 Oct 2013
Posts: 197
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, 23:08
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: 60627
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

25 Jan 2015, 03:43
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, 01:43
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: 86
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 02:35
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: 60627
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 02:49
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: 86
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 05:11
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: 60627
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

14 Jun 2017, 05:35
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);
_________________
Senior Manager
Status: Whatever it takes!
Joined: 10 Oct 2018
Posts: 382
GPA: 4
The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

18 Oct 2018, 01:00
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: 10007
Location: Pune, India
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

18 Oct 2018, 02:15
2
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

Non-Human User
Joined: 09 Sep 2013
Posts: 13999
Re: The expression x[n]y is defined for positive values of x and  [#permalink]

### Show Tags

04 Dec 2019, 01:58
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: The expression x[n]y is defined for positive values of x and   [#permalink] 04 Dec 2019, 01:58
Display posts from previous: Sort by