Question: Equivalent Powers

Comment on Equivalent Powers

x^n= y^n. Can we not do this: x= n√y^n n.n cancels out leaving x=y?
gmat-admin's picture

No, we can't do that for a few reasons.
First, if n = 0, then x^n = y^n for all values of x and y.
For example, although 2^0 = 1^0, we cannot conclude that 2 = 1

Second, if n is an even integer, then x^n = y^n does not necessarily mean that x = y.
For example, although 3^2 = (-3)^2, we cannot conclude that 3 = -3

This question by far is one of the best questions, if not the best. It looks simple, yet the question contains crucial points. I really appreciate your well thought-out question. Thanks.
gmat-admin's picture

Thanks Lee!

Brent, is it possible to have a rule that states that (a)^n =(-a)^n when n is even
gmat-admin's picture

That's a great rule!

We can even prove it.

(-a)^n = [(-1)(a)]^n = [(-1)^n][a^n]
= (1)[a^n] since -1 raised to an even power equals 1
= a^n

I did not get how case 3 make statement 1 sufficient.
Our rephrased question states n should be 0, in case 3. How did you determine x/y ^ n , and n=0?
gmat-admin's picture

We are told that x^n = y^n
There are 3 possible ways in which this equation can hold true.
In other words, at least one of the possible cases MUST be true:

case i) x/y = 1 (that is x = y)
case ii) x/y = -1 AND n is even
case iii) n = 0

Statement 1 allows us to rule out cases i and ii
This means case iii MUST be true (i.e., n MUST equal 0)

hey that was a great explanation Brent.
I came up with another way to solve it.

statement 1 says: x/y=2
lets suppose values for x and y
let x=4 , y=2
this satisfies statement one as 4/2=2
now put these values in given x^n=y^n
which makes 4^n=2^n
there is only one possible condition which can make this statement true,i.e when n=0
as 4^0=2^0=1
hence sufficient.

Statement 2 goes same as yours
please tell me if I am going wrong somewhere.

gmat-admin's picture

That's a valid solution.

I must say that rephrasing DS questions has been the most helpful tip I got from you on solving DS questions. I am learning by practice that the time it takes to rephrase saves me much more time I should have spent messing around interpreting both statements.
Thanks a lot!
gmat-admin's picture

I'm glad you like the tip.

Some people fail to recognize that, by devoting some time to rephrasing the target question, we can actually analyze the statements faster.

How did we know that case iii) n = 0 is the applicable related to the given information
Could anyone explain this part to me?
gmat-admin's picture

I'm not 100% sure what you're asking, but I have a feeling that you're wondering how we can be certain that n = 0 (in statement 1). If so, here's my response:

We're told that x^n = y^n

There are EXACTLY 3 possible ways in which this equation can hold true.

In other words, if x^n = y^n, then at least one of the following cases MUST be true:

case i) x/y = 1 (that is x = y)
case ii) x/y = -1 AND n is even
case iii) n = 0

Statement 1 allows us to rule out cases i and ii, leaving us with case iii (n = 0). So, it MUST be the case that n = 0

In statement 2, we're able to test some values to show that we cannot answer the target question with certainty.

Does that help?

Brent, great question. Made me realize the value in rephrasing the question stem.
While I understood this one off question, I have a feeling that without further practice of the same kind of question, it is likely going to fade away. For some reinforcement, where can I find similar type questions I.e. Deducing the question stem type
gmat-admin's picture

Yes, rephrasing the target question can be a huge time-saver for many Data Sufficiency questions.

If you want to see more instance where I've rephrased the target question, go to the Beat The GMAT forum or GMAT Club forum, and perform a search for "This is a good candidate for rephrasing the target question" (a preset phrase I often type when answering Data Sufficiency questions), you'll find all of the instances in which I've rephrased the target question.

I hope that helps.


Is 0 considered as an Integer as per GMAT?
gmat-admin's picture

Yes, according to the GMAT test-makers, zero is an integer.
Zero is also considered an integer by mathematicians, teachers and everyone else :-)


Is this the maximum level of Powers & Roots in GMAT or is it beyond 800 score?
Honestly my answer was wrong
gmat-admin's picture

I'd say this question falls in the 700-750 range of difficulty.


Hi Brent,

Can you please explain on the approach for solving this problem? There are some answers on gmatclub, but they skip steps making it hard to follow. Would appreciate it if you could give an overall tip on how to solve these questions and explain the logic behind the solution:

Thanks, much appreciated!
gmat-admin's picture

OK, somehow I just don't realize that if two integers are equal, one divided by the other one is 1
gmat-admin's picture

How did you conclude that x = y?

is x not equal to y here? therefore how is x/y = 2?
gmat-admin's picture

How did you conclude that x = y?

Office Hours

On December 20, 2023, Brent will stop offering office hours. 

Change Playback Speed

You have the option of watching the videos at various speeds (25% faster, 50% faster, etc). To change the playback speed, click the settings icon on the right side of the video status bar.

Have a question about this video?

Post your question in the Comment section below, and a GMAT expert will answer it as fast as humanly possible.

Free “Question of the Day” emails!