Know the GMAT Code: Exponents
How well do you know how to decipher the key clues that the test writers hide in GMAT problems? To get a really high score on this test, youve got to Know the Code in order to make your way through the toughest problems.
If this is your first introduction to the Know the Code idea, follow the link in that last paragraph to learn more. Then, come back here and practice your newfound skill on the problem below from the free GMATPrep exams.
Here you go:
*If [pmath]x[/pmath] > [pmath]y^2[/pmath] > [pmath]z^4[/pmath], which of the following statements could be true?I. [pmath]x[/pmath] > [pmath]y[/pmath] > [pmath]z[/pmath]
II. [pmath]z[/pmath] > [pmath]y[/pmath] > [pmath]x[/pmath]
III. [pmath]x[/pmath] > [pmath]z[/pmath] > [pmath]y[/pmath]
(A) I only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III
Ready? Lets do this!
Glance at the problem. PS. Roman numeral ugh. I may have to jump through a few more hoops than usual on this one. At this point, glance at the math. If it looks super hard and its a Roman numeral, then this might be a very good time to bail.
In fact, this problem can easily turn into a nightmare if you dont pick up on some Know the Code clues to help you make the solution process a bit more manageable. (Even then, this is still a hard one.)
Lets start getting this down on paper.
I double-checked everything after I wrote it down. Those statements are too close to each other; I dont want to get this wrong just because I made a mistake transcribing them.
Think about whats going on here (reflect!). Whats the best approach?
Whats the significance of the fact that [pmath]x[/pmath] is greater than [pmath]y^2[/pmath]? What needs to be true of these two numbers? And then the same question for the next bit: [pmath]y^2[/pmath] > [pmath]z^4[/pmath].
All right, now youre allowed to do some Work. :) Play around with some real numbers to understand the significance of the inequalities. (This is more formally called Testing Cases.)
Remember: youre only allowed to try numbers that make the given inequality, [pmath]x[/pmath] > [pmath]y^2[/pmath] > [pmath]z^4[/pmath], true. (Note: + int is my abbreviation for positive integer.)
First, I tried just [pmath]x[/pmath] and [pmath]y[/pmath] to make sure I understood what was happening. Then I added in [pmath]z[/pmath] and fully tested the problem. Statement I can be true. Take a look at the answer choices. Cross off answer (D).
Weve tried the easiest case: theyre all positive integers. What else could we try that might make statements II or III true?
Reflect again: heres where we need to Know the Code. Whenever you see an exponents Theory problem (like this one) where certain of the variables + exponents are greater than other ones, ask yourself: what are the weird cases when working with exponents?
Normally, when you square something, it gets larger. Thats true for all positive numbers greater than 1. Its also true for all negative numbers.
Its not true, though, for 0, 1, and fractions between 0 and 1. So heres where we may find cases in which statements II or III are true.
0 and 1 both stay the same when theyre raised to a power. Fractions between 0 and 1 get smaller. Glance at the statements: statement II is the reverse of statement I, which did work with positive integers. So maybe statement II will work with fractions between 0 and 1, which do the reverse of positive integers with respect to exponents?
Before, we made z the smallest integer, then [pmath]y[/pmath], and then [pmath]x[/pmath]. Can we reverse that trend if we use fractions instead?
It worked! Statement II could be true, too. Cross off answers (A) and (C). Now, were down to answers (B) and (E). Can we make statement III work?
This is going to be the tough one. Is there a way to get [pmath]z[/pmath]to jump in the middle, between [pmath]x[/pmath]and [pmath]y[/pmath]?
The first case used all positive integers and the second used all fractions between 0 and 1. To get the variables to work in the jumbled order of statement III, were probably going to have to jumble the characteristics of the numbers we try.
And we still, of course, can only try numbers that make [pmath]x[/pmath] > [pmath]y^2[/pmath] > [pmath]z^4[/pmath]true.
Hmm. How to make [pmath]x[/pmath]be the largest but have [pmath]z[/pmath]be greater than [pmath]y[/pmath]?
The borderline numbers are 0 and 1, so lets try putting the middle number, [pmath]y^2[/pmath], on a borderline number, [pmath]y=1[/pmath].
Now, [pmath]x[/pmath]has to be greater than 1. Lets call that 2. And [pmath]z^4[/pmath] has to be smaller than 1, so lets call [pmath]z=1/2[/pmath].
[pmath]x = 2[/pmath]
[pmath]y = 1[/pmath]
[pmath]z = 1/2[/pmath]
Whoops. That doesnt make [pmath]z[/pmath] greater than [pmath]y[/pmath].
What if [pmath]x[/pmath]is an integer but both [pmath]y[/pmath]and [pmath]z[/pmath]are fractions?
[pmath]x = 1[/pmath]
[pmath]y = 1/2[/pmath] (so[pmath]y^2 = 1/4[/pmath])
So[pmath]z^4[/pmath] would have to be smaller than [pmath]1/4[/pmath]even though z is greater than [pmath]y[/pmath]. Is that possible?
Sure! Try something just a bit larger than [pmath]y = 1/2[/pmath].
[pmath]z = 2/3[.pmath]
[pmath]z^4= 16 / 81[/pmath]
I have no idea what that fraction is in decimal form, but I know that [pmath]1/4[/pmath]is 0.25 and I know that [pmath]16/81[/pmath]has to be less than 0.25 because 0.25 would be about [pmath]20/80[/pmath].
To recap:
[pmath]x = 1[.pmath]
[pmath]y = 1/2[/pmath] (so[pmath]y^2 = 1/4[/pmath])
[pmath]z = 2/3[/pmath] (so[pmath]z^4= 16 / 81[/pmath])
[pmath]x[/pmath] > [pmath]y^2[/pmath] > [pmath]z^4[/pmath] Yes
III. [pmath]x[/pmath] > [pmath]z[/pmath] > [pmath]y[/pmath]True!
The correct answer is (E). Statements I, II, and III could all be true.
You could get sucked into trying a million different sets of numbers on this one and still not find your way to the precise type of numbers you need to make statement III work. We didnt even bother to try negative numbers on this one, even though normally Id think of negatives right after positives when testing cases. I didnt bother, though, because negatives do the same thing as positive integers (or positives greater than 1) when youre raising to even powers.
This is where the Code is so crucial: if you know that exponents get all weird when youre dealing with 0, 1, or fractions between 0 and 1, then youll have a much more targeted approach.
Key Takeaways for Knowing the Code:
(1) Use the clues in the problem to help you know what to try. These clues will always be there! You just have to learn how to read them.
(2) Here are the Know the Code takeaways that I came up with for this problem (make your own flash cards in your own words!):
* GMATPrep questions courtesy of the Graduate Management Admissions Council. Usage of this question does not imply endorsement by GMAC.