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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

Looking at it you can know that 2^k has a much greater growth rate than k^2 so based on this and the very quick growth of both functions you should start pluging in numbers with 0 and going up (as negative numbers squared can never equal anything raised to a negative number) you will find that it works for 1 and 2 and that at 4 2^k is much greater than k^2 so you can stop.

Easy, only two ways this can happen 2^2 = 2^2 and 2^4 = 4^2. C
_________________

I am the master of my fate. I am the captain of my soul. Please consider giving +1 Kudos if deserved!

DS - If negative answer only, still sufficient. No need to find exact solution. PS - Always look at the answers first CR - Read the question stem first, hunt for conclusion SC - Meaning first, Grammar second RC - Mentally connect paragraphs as you proceed. Short = 2min, Long = 3-4 min

Re: For how many integers k is k^2 = 2^k ? [#permalink]

Show Tags

05 Jan 2013, 19:45

Analytically it is not possible to see that these are the only 2 integer solutions. But you can find out that these are the ONLY possible solutions. For this, TRY GRAPHING

The best way to see that these are the only solutions is to look at the graph of these two functions: f(k) = kˆ2 is a parabola with vertex on (0,0) g(k) = 2ˆk is a exponential curve that has (0,1) as its y intersect

If you trace both graphs you'll see that the tail of the exponential curve does in fact cross the parabola for a value of k < 0 . However you can easily cross out the possibility of this being an integer value of k by testing out k = -1 and k = -2. (ps: Just for curiosity ,this value is a non rational value, aprox. -0.76)

For positive values, there are two intersections (that happen to be integers!). They are, in fact, the trivial k=2 and k=4 solutions most people guessed by testing numbers. It is not hard to see that the first intersection will have to happen, and given how the rate of increase for the exponential curve picks up much faster than the rate of increase for the quadratic function, they will have to cross paths once more. After that, the rate of increase for the quadratic does not catch up to the exponential, making it impossible for a third positive intersection.

Re: For how many integers k is k^2 = 2^k ? [#permalink]

Show Tags

11 Aug 2014, 07:30

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: For how many integers k is k^2 = 2^k ? [#permalink]

Show Tags

05 Jun 2016, 07:29

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.
_________________