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GRE Exponents: Practice Question Set

Each of the math questions below is directly inspired by a question in the on-line Revised GRE test. I’ve provided an easier version of the question (#1) and a more difficult version of the question (#2).

My recommendation is to try the easier version first. Then, if you answer it correctly, click on the link, and take a stab at the actual Revised GRE question.

If you are able to answer that question correctly, then as prize – you get a fiendishly difficult question (#3). Okay, maybe that’s not a prize – but it is great practice for those aiming for the 90% on quant.

The good news is I have explanations. For the Revised GRE question, I have recorded an explanation video you can watch. Finally, it is a good idea to try the easy question before the medium one, and the medium question before the difficult one.

Good luck!


1. Difficulty: Easy

If 0<10^n < 1,000,000, where n is a non-negative integer, what is the greatest value of 1/2 ^n?

  1. ½
  2. 1
  3. 5
  4. 32
  5. 64


Explanation: Don’t think big – think small. That is the smaller n becomes the greater ½^n becomes. So what is the smallest value? You may be tempted to say 1, which would give us ½. But remember n = 0, because 10^0 = 1. Therefore 1/2^0 = 1 Answer: B.

The “hidden zero,” as I like to call it, is a classic GRE math trick. So always keep your eyes open, especially when you see “non-negative integer,” which includes zero.


2. Difficulty: Medium-Difficult

(5/4^-n) < 16^-1

What is the least integer value of n?


The best place to start here is by getting rid of the unseemly negative signs and translating the equation as follows:

(4/5)^n < 1/16

A good little trick to learn using 4/5 taken to some power is that (4/5)^3 = 64/125, which is slightly—but only slightly—greater than ½. Therefore, we can translate (4/5)^3 to ½.

(1/2)^4 = 1/16

That would make (4/5)^12 a tad larger than 1/16. To make it less than 1/16 we would multiply by the final 4/5, giving us n = 13.



3. Difficulty: Hard

The equation n < 1/{(-2)^{-n}} < 135.43 is true for how many unique integer values of n, where n is a prime number?

  1. 7
  2. 4
  3. 2
  4. 1
  5. None of the above

This problem can be difficult, indeed downright inscrutable, unless you take your time and process one piece of information at a time. Once you understand what the problem is saying, you should be able to solve the question relatively quickly.



The most important piece of info is n is a prime number. So do not start by plugging in zero or one. Neither is a prime. The lowest prime is 2. When we plug in ‘2’ we get:

2 < 1/{(-2)^{-2}} < 135.43

2 < 4 < 135.43

This is clearly true. Thus we have one instance.


As soon as we plug in other prime numbers a pattern emerges.

1/{(-2)^{-n}} is always a negative number if n is odd. Because all of the primes greater than 2 are odd, the number in the middle will always be negative:



Because in each case n is a positive number we can never have the middle of the dual inequality be positive, if n is an odd prime.

Thus the only instance in which the inequality holds true is if we plug in ‘2’, the answer is (D).

If you got that right – congratulate yourself. It’s a toughie.


By the way, students who use Magoosh GRE improve their scores by an average of 8 points on the new scale (150 points on the old scale.) Click here to learn more.

40 Responses to GRE Exponents: Practice Question Set

  1. Allie July 21, 2016 at 3:40 pm #

    Hi Chris,
    For question 2, is there another way to solve it? I read through the comments but I don’t get how you got to (4/5)^12.
    Could we possibly just sub in different values for n until we get to 13?

    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert July 24, 2016 at 2:26 pm #

      First, you could definitely sub different values for n until you arrive at (4/5)^13. However, this would require a lot of time consuming trial and error, and time is never on your side in GRE Quants.

      Instead, it’s better to use shortcuts and estimation wherever possible, to get to the answer more quickly. This is the approach Chris took to the second, medium-hard level problem in this post.

      I just re-read the comments myself, and I can understand why the comments alone might not be enough for you to understand the shortcuts Chris took. There hasn’t yet been an explanation of how this problem is solved, from the very first step to the very last one. But I can fix that! 🙂

      So here are the steps in detail (and let me know if you still have questions after this, Allie).

      STEP 1: Get rid of the negative signs in the exponential notations. To keep the numeric value of each expression the same, if you get rid of the negative sign in the exponents, you have to reverse the position of the numerators and denominators, “flipping” each fraction. And so, (5/4)^-n < 16^-1 becomes (4/5)^n <(1/16)^1, or simply (4/5)^n < 1/16.

      STEP 2: Use “number sense” (the ability to make hidden but relatively simple connections between numbers) to recognize that (4/5)^3 is 64/125, which is slightly more than 1/2.

      STEP 3: Recognize that (1/2)^4 = 1/16, the lesser number in the inequality. This is also a matter of number sense, but it’s an easy thing to spot with a basic sense of numbers. 1^(anything) is still 1, and 2^4 is 16. You should be able to make this connection just by looking at 1/2 and 1/16 and recognizing that (1/2)^4 becomes 1/16. From there, you can realize that if (1/2)^4 = 16, and (4/5)^3 is slightly greater than 1/2, than (4/5)^3 must also be slightly greater than 1/16.

      STEP 4: Safely guess that if (4/5)^3 is a little greater than 1/2, and (1/2)^4 is equal to 1/16, then (4/5)^(3*4)— or (45)6^12— is slightly greater than 1/16. Why do you take the ^4 from 1/2, and multiply it by the ^3 after 4/5? Well, remember that 1/2 is a modified, smaller approximation of the value of (4/5)^3. To get the real exponential value of 4/5 that slightly exceeds 1/16, you need to turn (1/2) back into (4/5)^3, again, and then add its ^4 exponent back in. The process looks like this: (1/2)^4 ===> [(4/5)^3]^4. And [(4/5)^3]^4 can also be expressed as (4/5)^3*4, or (4/5)^12.

      STEP 5: Now you have an exponential expression that is just a tiny bit bigger than 1/16. And the root number of the exponent, 4/5, is less than 1. Anytime you exponentially multiply a number smaller than 1, it goes down in value. So if (4/5)^12 is slightly greater than 1/16, then (4/5)^13 will be slightly smaller than 1/16. This makes 13 the lowest value you can plug in for n in the original inequality.

      These five steps may look complicated, as I’ve explained them in as much detail as possible. But if you can get the hang of this kind of estimation and number sense, approaching this problem with the steps above will be faster than simply using trial and error to arrive at 13 as the correct value for n.

      Again, let me know if you have any more questions or doubts about this problem and how to solve it. 🙂

  2. Adarsh June 24, 2016 at 7:30 pm #

    hi chris ,

    In question .2 ,Is it (5/4^-n) or (5/4)^-n


    • Magoosh Test Prep Expert
      Magoosh Test Prep Expert June 29, 2016 at 2:18 am #

      Hi Adarsh 🙂

      In question 2, the left side of the equation is (5/4)^(-n). To rewrite this term with a positive exponent, we need to “flip” the fraction, which leaves us with the reciprocal of the original term:

      (5/4)^(-n) = (4/5)^(n)

      Hope this helps 🙂

      • Adarsh June 30, 2016 at 12:20 am #

        hi there ,
        yeah i got that but it was written (5/4^-n).

        • Magoosh Test Prep Expert
          Magoosh Test Prep Expert July 2, 2016 at 5:58 am #

          Hi Adarsh 🙂 Happy to have helped! I included the parenthesis to emphasize what the expression was saying. Negative exponents or exponents with more than one term can look confusing based on the formatting of the exponent. On the actual exam, all exponents will be in superscript (and not expressed using the “^” symbol), so it should always be clear what exponent the base is being raised to 🙂

  3. Mel October 20, 2015 at 10:49 am #

    I still don’t understand number 1. Can you please explain using other words?

  4. Ji September 10, 2015 at 1:03 am #

    Hi Chris,

    Would you be okay if you can explain why the answer is 13 for question #2, please?
    So far, I understand (4/5)^n < 1/16 but from that point, I lost why the answer is 13.

    Please let me know and I thank you in advance!:)

    • Chris Lele
      Chris Lele September 10, 2015 at 10:59 am #

      Hi Ji,

      Sure, I’d definitely be happy to explain! This is quite a toughie 🙂

      So, what I did with 4/5 is I realized that by making n = 3, I get 64/125. That is very close to 64/128. So though it’s a tad larger, 64/125 for convenience sake can be simplified to 1/2. At this point the problem can be rewritten, using a new variable for an exponent, r.

      1/2^r = 1/16

      In this case r would equal 4. Since (4/5)^3 = 1/2, we can say (4/5)^3(4) or 4/5^12 = 1/16. The thing is it doesn’t quite equal 1/16 since (4/5)^3 is slightly larger than 1/2. Therefore, (4/5)^12 will be slightly larger than 1/16. By increasing 12 by 1, to get 13, we will definitively make the left side of the equation smaller than 1/16.

      Hope that helps!

      • Ji September 11, 2015 at 2:53 am #

        Aha! Thanks to your step-by-step explanation, now it does make sense to me. However, I am pretty concerned that I won’t be able to get this type of question right on the actual test. Persistently to practice makes me perfect, I guess. 🙂

        Thank you Chris! 🙂

  5. Meghna June 27, 2015 at 11:34 pm #

    I have a question regarding the following method of solving Q3:

    I basically converted the equation into:

    n < (-2)^n < 135.43

    After which I assumed n to be a positive prime leaving only 1 possible prime, ie. 2.

    Is this alright? Or is there a flaw in this reasoning?
    Thank you.

  6. Samy October 13, 2014 at 9:49 am #

    Please update the link for Question no. 2 above in the blog (medium difficulty problem) with the correct link posted by respondent/Magoosh member Chris on April 25, 2014 at 9:25 pm. The link posted by Chris lele or Magoosh is actually incorrect. Thanks for posting this problem.

    • Chris Lele
      Chris Lele October 13, 2014 at 11:23 am #

      Hi Samy,

      That link seems to have disappeared :(. No worries–I wrote an entirely new question for the second question in the three above. Hope you like it!

      • Laz April 17, 2015 at 8:17 am #

        <a (5/4^-n) < 16^-1

        Could you please clarify question 2 above ? I don't see how a has to do with anything ?

        • Chris Lele
          Chris Lele April 27, 2015 at 4:59 pm #

          Hi Laz,

          Not sure how that got in there–sorry for any confusion! I’ve removed the “<a".

  7. Muneeza August 23, 2014 at 12:47 am #

    This blog is super helpful! However, I live in a country where YouTube is banned so unfortunately I cant take advantage of the video explanations. Can you please at least tell me the answer to the ‘medium difficulty’ question. Thanks, I really appreciate it!

    • Chris Lele
      Chris Lele August 25, 2014 at 3:29 pm #

      No problem 🙂

      The answer is (A).

      Hope that helps!

  8. Hanna January 2, 2014 at 6:39 pm #

    Hi Chris,

    For your second question, I don’t think the link goes to the correct question. Section 5 question 12 is a word problem about floor space. Could you please provide the correct question?



  9. Arpita August 25, 2012 at 5:07 pm #

    Hi Chris,

    Is that still the correct link for #2? Because when I click it and go to section 5, #12, it’s some rectangular area problem. Thanks!

    • Chris Lele
      Chris August 27, 2012 at 12:14 pm #

      Hmm…it’s still working for me…try it again, but if it doesn’t work let me know and I’ll try to figure something out :).

      • Axl Laruse April 21, 2013 at 8:37 am #


        I think Arpita is saying that at the section 5, the problem #12 does not match with the video you are posting.

        On the page #76 there are no exponent problems that match your video, the same goes from the page #74 to #81; therefore, the post have the wrong page, section and problem number.

        • Chris Lele
          Chris Lele April 22, 2013 at 12:33 pm #

          Hi Axl,

          Hmm…that is strange…well, I can’t seem to find the problem either now. I will replace it with another exponent problem.

          Thanks for point that out :)!

  10. Shree June 29, 2012 at 8:10 am #

    *just need a confirmation

  11. Shree June 29, 2012 at 8:09 am #

    The important thing i learned over here is that prime numbers cannot be negative integers, because if you put n=-2,-3,-5 etc, the equation above is satisfied for an infinite number of values, that’s why i answered “none of the above”

    just as a confirmation, can negative integers be prime numbers?

    • Chris Lele
      Chris June 29, 2012 at 2:47 pm #

      Hi Shree,

      Prime numbers are only positive integers. Negative integers cannot be prime numbers.

      Hope that helps!

  12. Dineshvalmiki May 21, 2012 at 2:00 pm #

    Everything is perfect just wanted to verify this below statement !

    “Thus the only instance in which the inequality holds true is if we plug in ‘2’, the answer is (D).”

    The answer is (C) i.e. 2 isn’t it ?

    • Chris Lele
      Chris May 21, 2012 at 3:13 pm #

      Actually, this is quite a tricky question :).

      ‘2’ is the only integer value that satisfies the equation. Thus there is a total of one integer value (the number ‘2’) that satisfies the equation.

      Hope that makes more sense :).

    • ankit June 19, 2013 at 3:36 pm #

      thanks.. I was also confounded!

  13. nani April 25, 2012 at 1:04 am #

    Hi Chris,

    I am just coming with lot of questions, i hope u may not get frustrated! :).

    The link you gave for second question in the ETS web, when i am searching for that answer i found there is an extra column called (P+), what is that P+ ?


    • Chris Lele
      Chris April 25, 2012 at 11:55 am #

      (P+) stands for the percentage of people who answered the question correctly. For instance, a difficult question only 25% or less of people get right. This can help you get a sense of the type of questions you are missing (meaning you be answering the easy ones correctly, and don’t get despaired if you miss the tough ones).

      Hope that helps, and don’t hesitate to ask any more questions :).

  14. Aman April 14, 2012 at 6:00 am #

    Got all right ,last question logic as you said is 2 is only prime no which is even and we want that value of n which can make (-2)^n =+ve…
    Good collection of mazed questions…

    • Chris Lele
      Chris April 16, 2012 at 11:40 am #

      Thanks! There will be some more tough math questions coming soon :).

  15. Syam February 27, 2012 at 10:45 pm #

    Oops! Prime numbers start from 2. How dumb of me 😛

    • Chris Lele
      Chris February 28, 2012 at 12:18 pm #

      Hi Syam, no worries 🙂

  16. Syam February 27, 2012 at 10:44 pm #


    Wouldnt ‘-2’ qualify as ‘unique’ integer? Does ‘unique’ imply that the absolute value should be unique?

  17. Zainab February 11, 2012 at 12:29 pm #


    You said that “Thus the only instance in which the inequality holds true is if we plug in ‘2’, the answer is (D).” The answer is C then?


    • Chris Lele
      Chris February 13, 2012 at 12:05 pm #

      Hey Zainab,

      This is a tricky question – it asks for the number of instances. There is only one instance – when you plugged in the number 2.

  18. Peng January 31, 2012 at 4:44 pm #

    Hi Chris,

    Thank you for the post.

    I found the last question very confusing the way -2^-n was written. How are you able to tell if it is -(2^-n) or (-2)^-n? Also, when there is no parenthesis displayed, wouldn’t you assume it is the former case since exponent has more priority?


    • Chris Lele
      Chris February 1, 2012 at 1:41 pm #

      Hi Peng,

      You are right – that is ambiguous. The negative numbers should be in parentheses, e.g. (-2)^-2

      Thanks for catching that!

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