# Lesson: Introduction to Exponents

## Comment on Introduction to Exponents

### Hey Brent,

Hey Brent,

regarding this Q with exponents and inequalities:

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

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? 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?

Cheers,
Brent

### I assumed two cases for the

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? ### Sorry, but I'm still not 100%

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,

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? ### You're right; Math Revolution

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 :)

Got it. Thanks a ton :)

### Hi Brent,

Hi Brent,

Can you please solve the below?

https://gmatclub.com/forum/p-and-q-are-prime-numbers-less-than-70-what-is-the-units-digit-of-p-q-196022.html#p1514019

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

Thanks ### Hi Brent,

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?

Thanks ### We're told P is a prime

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.

### https://gmatclub.com/forum/if

https://gmatclub.com/forum/if-b-is-an-integer-greater-than-1-ab-222195.html

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... ### The good thing is that you

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

### For the reinforcement

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

https://gmatclub.com/forum/if-x-y-2-z-4-which-of-the-following-statements-could-be-100465.html

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 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

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? ### As you can imagine, the more

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.