Lesson: Introduction to Exponents

Comment on Introduction to Exponents

Hey Brent,

regarding this Q with exponents and inequalities:


In the two statements, since I don´t know the sign I can´t simply divide or multiply. But can I do si if I consider the case of x being positive and x being neg.? It worked out for me on this one.

To be more specific:

Stmt 1: 1/x > -1

case 1 (x pos): -1<x
case 2 (x neg): 1<x

thus, insuff.

Stmt 2: 1/x^5>1/x^3

case 1 (x pos): 1>x
case 2 (x neg.): 1>x

thus, suff.

Is that possible? or did it luckily work out with the specific set of numbers in that particular Q?

gmat-admin's picture

Question link: https://gmatclub.com/forum/gmat-diagnostic-test-question-79337.html

Without seeing your full solution, it's hard to tell whether the steps you took are valid.

That is, for Statement 2 you write:
case 1 (x pos): 1 > x
case 2 (x neg.): 1 > x
These are correct conclusions (for each case), but how did you arrive at those conclusions?


I assumed two cases for the statement: Case 1 x being negative, so when I multiplied by x I switched the < and the other case assuming x being positive, so I miltiplied but didn´t switch the <.

However, I just noticed that when doing that for stmt 2 (which actually yields an answer), I get for assuming x being neg. and pos. respectively two contradicting results. So I guess my approach is invalid?
gmat-admin's picture

Sorry, but I'm still not 100% clear on your solution.
For statement 2, are you multiplying both sides by x or x^3 or x^5?
In the meantime, Bunuel provides a nice solution here: https://gmatclub.com/forum/gmat-diagnostic-test-question-79337-20.html#p...

Hi Brent,
Bunuel and Math Revolution have solved this question completely differently and reached different ranges of 'x'. And I think Math Revolution's approach is not correct here. What do you think?
gmat-admin's picture

You're right; Math Revolution's approach is not correct for statement 2.
The problem with MR's solution is that they take 1/x^5 > 1/x^3, and multiply both sides of the inequality by x^5 to get 1 > x^2.
However, if x is negative, then x^5 is also negative, which means we must reverse the direction of the inequality sign.

Here's my approach to statement 2: 1/x^5 > 1/x^3
Since x ≠ 0, we know that x^6 must be positive.
So we can safely multiply both sides of the inequality by x^6 to get: x > x^3
Rearrange to get: x^3 - x < 0
Factor: (x-1)(x)(x+1) < 0
From this we can conclude that x < -1 OR 0 < x < 1 (the same as Bunuel's range of values)

Got it. Thanks a ton :)

Hi Brent,

Can you please solve the below?


In the first statement, why would the units digit of Q be 7?

gmat-admin's picture

Hi Brent,

Thanks for the detailed solution. However, how did you come up with this one for statement 2:

So, if P^(4k+2) = 729?

gmat-admin's picture

We're told P is a prime number, and that k is a positive integer.
So, I tested P = 3 and k = 1, to get: P^(4k+2) = 3^[4(1)+2] = 3^6 = 729.


How could I completely forgot any number times 0 is 0...

This is just stupid, I chose C because I know a but don't know b, this is a big mistake I need to take note of...
gmat-admin's picture

The good thing is that you made this mistake while practicing (and not on test day!)

For the reinforcement questions over 500, I got nothing right because I always ignore information or over interpret things, for example


is very abstract, more complicated, what I initially think is just wrong, I think x MUST be greater than y and z thus the answer should be I and III
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-x-y-2-z-4-which-of-the-following-statement...

Tricky question!! (only 54% answered correctly)
This is a good question because it hinges on the difference between MUST be true and COULD be true.

If x = 1/4, y = 1/3, and z = 1/2, then x > y² > z⁴
So, statement II COULD be true.

Hi Brent
I try to do all the reinforcement questions given with each video but they just seem to be too many, and even if I filter just the 500-650+ ones, I noticed that there are some questions with 60+ difficulty in the <500 ones too.
I think I will have to be selective if I wish to complete all videos+OG questions as well. How should I plan it?
gmat-admin's picture

As you can imagine, the more practice questions you can answer, the better your score. However, not everyone has the time to answer 4000+ practice questions!

For starters, I suggest that you answer at least 3 or 4 practice questions from the Reinforcement Activities box beneath each lesson. Be sure to answer questions with a difficulty level similar to your target score. For example, if you're aiming for a medium quantitative score, but then be sure to answer questions in the 500-650 range.

Also keep in mind that the number of questions in a particular Reinforcement Activities box is directly proportional to how frequently that particular topic is tested. For example, inequality questions are very popular on the GMAT, and you'll find TONS a practice questions under that particular lesson (https://www.gmatprepnow.com/module/gmat-algebra-and-equation-solving/vid...). So, if you see a ton of practice questions under a certain topic, be sure to answer enough practice questions so that you master that topic.

How do the Reinforcement Activities work?

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