Welcome back! Today, we’ll go over Tuesday’s Brain Twister problem.

## Question

x^n + x^n + x^n = x^(n + 1), where x cannot equal zero.

Which of the following must be true?

I. n = 2

II. x = 3

III. n^x < x^n

(A) I only

(B) II and III

(C) I and III

(D) II only

(E) None of the above

## Answer and Explanation

There are a few ways to flub this question. One is to think that there is no way to know the value of ‘n’ or ‘x’ and then to blithely mark (E) as the answer. The reality is that one of the variables is a known, based on the relationship between the two sides of the equation.

Notice how there are three ‘x^n’ values on the left and on the right side of the equation we have x^(n + 1). Therefore, ‘x’ has to be ‘3’, since we are adding one more ‘x’ (remember the (n+1)).

Another way to miss this question is by forgetting the case for where x could equal n. So if we know that ‘x’ has to be three and we make n = 3, then Condition III is not true.

Therefore, the answer is (D).

See you again soon!

Sorry, why is condition III (n^x<x^n) not correct?

I’ll be happy to explain that in greater detail. Basically, there are cases where condition III

canbe correct, and cases where condition III isnottrue. And a statement only MUST be true if it’s true under all conditions.x must equal 3. So Statement II must be true. n can equal something other than 2, so statement I is not true all the time, and therefor you can’t say that it must be true, only that it can be true.

Now, if statement 1 is sometimes true and sometimes not, then statement 3, which says that n^x < x^n would also have to be sometimes true, sometimes not. Let's look at a case where condition III is true. Let's say that n really is 2, while x is what it always must be: 3. We'll plug in the numbers:n^x < x^n 2^3 < 3^2 8 < 9 TRUEBut what if n is 3, and x is also 3? This is a possibility, based on the original formula in the initial prompt, the one that reads: x^n + x^n + x^n = x^(n + 1). Well, let's run the numbers:n^x < x^n 3^3 < 3^3 NOT TRUE Actually, 3^3 = 3^3So you see, condition III can be true or untrue. So we can't say it absolutely

must be true.Daniel November 27, 2015 at 10:52 am #Hi Chris,What about if x equals zero and n equals (for instance) 1? Then the equation is satisfied: on the left side, each of the terms equals zero (0 to the 1st power), so their sum equals the term on the right (0 to the 2nd power). Since the equation has a solution where x does not equal 3, answer choice D must be incorrect and answer choice E must be correct. Do you see any problems with my reasoning here?Vishnu June 9, 2015 at 11:59 pm #So, we get x=3 and n= 1 . Am I correct ?Joshua June 2, 2015 at 12:33 pm #Hi,Your answer should be D, II only, right?

Thanks!

Chris Lele June 2, 2015 at 5:16 pm #Yes, my answer was supposed to be (D). I made a typo :).Ishaq June 1, 2015 at 3:17 pm #don’t you mean the correct answer is D?Priyansh Jain June 1, 2015 at 10:48 am #The answer should be ‘D’ as condition 3 only holds true.Vamshi June 1, 2015 at 4:34 am #Answer is D.Manjinder Singh May 31, 2015 at 11:36 pm #what if ‘x’ equals zero.or can you please explain why ‘x’ can not be equal to zero.Vishal May 31, 2015 at 11:01 am #Chris, The answer can’t be A.If the answer is A,

then it will be evaluated to 3 * x ^2 = x ^ (3)

If x is valued as 1, there is inequalities on the both side.Therefore, It can’t be A.Chris Lele June 2, 2015 at 5:17 pm #Yes, that was a typo on my part :). I meant (D), not (A).Carlos May 31, 2015 at 5:44 am #“Therefore, the answer is (A).”Sorry, I meant the answer is D.Chris Lele June 2, 2015 at 5:18 pm #Yes, yes :)…that was a typo on my part.VInod May 30, 2015 at 9:18 am #A? o.O. Lele, Is it a typo or am I missing something important in this seemingly easy question?I rather hope not!

Chris Lele June 2, 2015 at 5:17 pm #Hi Vinod,No, you are not missing anything :). And yes, that was a typo 🙂H-MAN May 30, 2015 at 8:36 am #Hi Chris,I think answer should be (D)Chris Lele June 2, 2015 at 5:17 pm #Yes, it should be (D). That was a typo 🙂Alankritha May 30, 2015 at 3:07 am #Hii..I don’t understand how u could conclude n=2 when the question is 3*x^n= x^(n+1) if n=2 exponent on lhs is 2 and on rhs is 3

ridhi May 29, 2015 at 11:53 pm #Hey Chris,I solved till x=3 and marked answer D as the correct one. However, per your explanation there is one more condition x=n. I’m totally lost how this condition is considered in this problem. Could you please elaborate ?Regards,Ridhi

Chris Lele June 2, 2015 at 5:21 pm #Sorry about that! Rereading my explanation, I too am initially confused. I will rework that explanation so that it makes more sense!Abhinav May 29, 2015 at 6:45 pm #Hi Chris,In the explanation you have mentioned that x has to be 3 in order to satisfy the equation. Thus, answer should be option D not A.

Chris Lele June 2, 2015 at 5:21 pm #Yes, that was a typo 🙂Azalea May 29, 2015 at 1:54 pm #Hey 🙂 Shouldn’t the answer be D?Chris Lele June 2, 2015 at 5:22 pm #Yep, that was a typo on my part 🙂