Infinite exponentials

Level pending

The field of infinite fractions and exponentials is incredibly interesting. Give this problem a shot:

$x^{x^{x^{...}}} = n$

What is $x$ ?

n^{1/n}

n\sqrt{n}

n 0 1

n^{1/2}

1 solution

Imran Qureshi
Mar 28, 2016

The problem looks nigh impossible if you try to keep thinking of numbers that would equal n if kept "exponentiating it". Instead let's think of the problem in a more clever way, consider it without the leading x on the left side, what do we have left?

$x^{x^{...}} = n$

Well that's pretty much the same as our previous equation, so let's plug that into our original problem:

$x^{n} = n$

Now its incredibly easy, just take the nth root of both sides - netting you $x = n^{1/n}$ which is our answer.

Infinite exponentials

1 solution

0 pending reports