If y^k/y^5k = m, what is k?

Question

If  \frac{y^k}{y^{5k}} = y^m , what is k?
 
(1) m = 4
(2) y = 4

BrentGMATPrepNow · Accepted Answer

Great question, Bunuel!!

Target question :    What is k?

Given :  \frac{y^k}{y^{5k}} = y^m   
IMPORTANT: many people will mistakenly apply some exponent laws and conclude that -4k = m. HOWEVER, before we apply these exponent laws, we must be certain that the base does not equal 0, 1 or -1, in which case the exponent laws fly out the window.

Statement 1 : m = 4   
Let's TEST some values.

There are several values of y, k and m that satisfy statement 1. Here are two: 
 Case a : y = 1, k = 1 and m = 4. Plugging this into our given equation, we get  \frac{1^1}{1^5} = 1^4 , which works! In this case,  k = 1

Case b : y = 1, k = 2 and m = 4. Plugging this into our given equation, we get  \frac{1^2}{1^{10}} = 1^4 , which works! In this case,  k = 2

Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : y = 4  
There are several values of x and y that satisfy statement 2. Here are two: 
 Case a : y = 4, k = 1 and m = -4. Plugging this into our given equation, we get  \frac{4^1}{4^5} = 4^{-4} , which works! In this case,  k = 1

Case b : y = 4, k = 2 and m = -8. Plugging this into our given equation, we get  \frac{4^2}{4^{10}} = 4^{-8} , which works! In this case,  k = 2

Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Since statement 2 tells us that y does NOT equal 0, 1 or -1, we can safely apply some exponent laws: 
Given:  \frac{y^k}{y^{5k}} = y^m 
Simplify left side to get:  y^{-4k} = y^m 
So, -4k = m
Statement 1 tells us that m = 4
Plug this into our equation to get: -4k = 4. which means  k must equal -1 
Since we can answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer:  C

RELATED VIDEO
 https://www.youtube.com/watch?v=LHz74_lvLI8

chetan2u · Answer

Hi..
All the terms are with base y and powers are k and m...
So m and k can be equated, requiring only value of m to find k..

\frac{y^k}{y^{5k}}=y^m........y^{k-5k}=y^m.....-4k=m.....k=-\frac{m}{4}

Let's see the statements
1) m=4
We will get the value of k
Sufficient

2) y=4
Nothing about k..
Insufficient

A

chetan2u · Answer

Hi...

I would differ on the coloured portion.
 The Q is telling us that the equality is True. So if you have y as 0, the equation becomes undefined and the very basis of the Q falls flat..
Similarly for other values.
If I say - " if y/y =y, what is y? I CAN NOT take y as 0
What you have conveyed is surely TRUE of the Q asked :- " IS y^k/y^5k=y^m ..
So value of m is sufficient to tell about value of k.

BrentGMATPrepNow · Answer

The part about the base not equaling zero is a general comment. 
For example, if we know that 0^x = 0^3, we can't conclude that x = 3

I don't agree your comments in blue. 
All we know is that the equation must hold true. 
My counterexamples for statement 1 satisfy the equation AND yield different answers to the target question. So, statement 1 cannot be sufficient.

Cheers,
Brent

septwibowo · Answer

Wow

Intereseting, GMATPrepNow , I think we forget these exceptions. Thanks for remind us!

chetan2u · Answer

hi...

I do not have any doubt on the logic as I would surely use it when it would have said - IS ... = ...

I am sure I have seen few Qs earlier where the final equation boils down to something like  y^2=y^k ..
so if someone tells me that - If  y^2+3y - (2y+2) = y^k +y-2 , what is k?
the equation boils down to  y^2=y^k .. here answer should be k=2 but the logic of y as 0, 1 or -1 will not give k as 2
I may have not come across the y as 0,1 etc in such situations, and it may actually be true and I may be wrong.
 so would request if you have come across such use in actuals, it will surely help many who would go through this thread

But if someone tells me - Is  y^2+3y - (2y+2) = y^k +y-2......y^2=y^k .. we cannot answer as all those values of 1, -1 or 0 come in.

BrentGMATPrepNow · Answer

I'm not sure I understand what you mean by the "IS" distinction. 
The target question asks "What IS the value of k"

ASIDE: If the question were...
What is the value of k? 
(1) 1^3 = 1^k
...would statement 1 be sufficient?

chetan2u · Answer

Hi...

i meant by "IS....=.." :-  Is   x^7=x^k ?
here I would surely check x for 0,1,-1 and other values

But if the Q says it -  IF   x^7+x^2+4x+4=x^k+(x+2)^2 , what is the value of k?
this will boil down to  x^7=x^k  and we should get k as 7..

1^3=1^k is a different matter as it is giving base as 1, but if there is a variable as x above, I have not seen a Q claiming it to be insufficient and substituting x as 0,1, and -1 .
as I said earlier, maybe I have not seen a Q, it could be that there are such Q in actuals. If so, it will help.

rahul16singh28 · Answer

Hi  Bunuel

This seems to be Debatable OA. Can you please share your views on this.

Bunuel · Answer

OA is not debatable, it's C. Check the solution here: https://gmatclub.com/forum/if-y-k-y-5k- ... l#p1919376

niks18 · Answer

\frac{y^k}{y^{5k}} = y^m=>y^{-4k}=y^m .

Statement 1:  Now here the base  y  is a variable and not a constant, hence   WE CANNOT EQUATE THE POWERS

For e.g if we had  5^{-4k}=5^m , then there powers could be equated i.e.  -4k=m , hence by knowing the value of  m , we could know the value of  k

But here if  y=0  or  1 , then irrespective of the value of  m ,  k  could take any value. For e.g  1^{-8}=1^{100} , so here  m=100  and  k  could be  2 .

Hence Statement 1 is  Insufficient

Statement 2:  nothing mentioned about  k  or  m .  Insufficient

Combining 1 & 2:  we know the value of  y  &  m , hence value of  k  can be calculated.  Sufficient

Option  C

HAPPYatHARVARD · Answer

For y=1, we can't find the value of k or m. so y has to be something other than 1 or zero. Hence option B is also necessary.

nick1816 · Answer

Apply log both sides
-4k log(y) = m log(y)
m= (-4k), unless y is equal to 0 or 1
Statement 1
m=4 byt we dont know whether y=0 or 1
insufficient
Statement 2
y=4
we don't know the value of m
insufficient
Combining both equations
y=4 and m=4
sufficient

Basshead · Answer

Are there any official problems that test the -1, 0, 1 in the base trap when dealing with exponents? Would love additional practice.

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If y^k/y^5k = m, what is k?

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