M32-14

Question

If  x>0  and the expression  x^{x^{x^{...}}} , where the given expression extends to an infinite number of exponents, equals 2, then what is the value of  x ?

A.  \frac{1}{2}

B.  \sqrt {2}

C.  \sqrt{2}

D.  \sqrt{3}

E.  2

Bunuel · Accepted Answer

Official Solution:

If  x>0  and the expression  x^{x^{x^{...}}} , where the given expression extends to an infinite number of exponents, equals 2, then what is the value of  x ?

A.  \frac{1}{2}

B.  \sqrt {2}

C.  \sqrt{2}

D.  \sqrt{3}

E.  2

Re-write as:  x^{(x^{x^{...}})} . Given that the expression extends to an infinite number of exponents, the expression inside the parentheses would also equal 2. This allows us to substitute the value inside the parentheses with 2 and rewrite the given expression as  x^2=2 . Hence,  x = \sqrt{2} .

Answer: C

Xavipersonal · Answer

I don't really understand the following  Since the expression extends to an infinite number of exponents, then the expression in brackets would also equal to 2.

How can the product of infinite x be equal to 2?

susheelh · Answer

Hello  Bunuel ,

I too have the same question as  Xavipersonal . How does the expression in bracker of  x^{(x^{x^{...}})}  end up as 2? Is this some kind of a rule?

Thanks in advance for your answer!

Bunuel · Answer

Let me ask you: does the expression in brackets differ from the expression which equals to 2 in any way? Similar questions: https://gmatclub.com/forum/if-the-expre ... 98647.html https://gmatclub.com/forum/new-tough-an ... l#p1029228 https://gmatclub.com/forum/find-the-val ... 38049.html https://gmatclub.com/forum/find-the-val ... 75403.html

coreyander · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

sssdd1234 · Answer

I think this is a  high-quality  question and I agree with explanation.

avohra223 · Answer

If x^(some number) = 2, then x has to be sqroot(2) since the only way to achieve the value "2" through exponents is Sqroot(2) raised to the power 2. So, x = sqroot2 and the exponent they have defined equates to 2.

pyreddy · Answer

Given that the expression extends to an infinite number of exponents, the expression inside the parentheses would also equal 2. Why it is equal to 2. ? Can you explain please

Bunuel · Answer

adarsh2209 · Answer

I like the solution - it’s helpful.

IJIJ · Answer

I like the solution - it’s helpful.

BlackHat098 · Answer

Hi BB,

From the threads on similar questions, I can gauge that infinitely recursive expressions are not tested on GMAT. But this was mentioned ~5 years ago, could you pls tell us what's the latest on this. Do these still form a part of the syllabus?

Bunuel · Answer

I cannot point you to any official GMAT question that explicitly tests infinitely recursive expressions. However, the underlying algebraic manipulations involved are very much within the scope of the exam. For that reason, this type of question is still useful for practice, even if not for anything more.

sequimaxime · Answer

For those who are asking for steps or logic behind the question lemme explain how we can simply arrive at the answer:
so lets put ,
y = x^(infinite x's)
if we carefully see , we can even write this as
y=x^(y) -----Our logic is still valid in here
so y=x^y=2
y=2 and also now x^2=2 
Hence ,
 X is root 2.

commodimaiores · Answer

I did not quite understand the solution. please specify in detail.

Bunuel · Answer

minuscorporis · Answer

This is a great question that’s helpful for learning. I think this is a high-quality question and I agree with explanation.

VenkataSai · Answer

Hi Bunuel,

I tried the below method, 
Given x^x^x^..= 2.--> eq1
Applying 1/x root on both sides. Then equation becomes
x^x^x..=2^1/x. -->eq2.
Substituting eq1 in eq2
2=2^1/x.

Hence x=1.

I know the answer is wrong, but can you mathematically explain why it is wrong.

M32-14

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