Expofinity

Consider the equation
x x x x x x x x x x · · ·= 3.

Can you solve it? Can you check your answer?

Like a lot of interesting algebra problems (if you're willing to call this algebra (I am)), there are a few approaches to solving the equation. All of the approaches that spring immediately to mind lead to the same result, so here is what I believe to be the simplest of them:

According to the equation itself, the entire exponent on x on the left side of the equation is equal to 3. Therefore x3 = 3, so x = ∛3 ≈ 1.44225.

Logically, it would seem that this result is the only possible solution to the equation, unless the unusual notation of the equation has some very unintuitive meaning. Can there be more than one reasonable interpretation of the equation? If not, does the above value of x actually solve the equation? That is, how do you "check" the potential solution x = ∛3? You may find that answering these questions will take you on a fun little journey.

Notes
  • I originally encountered this equation written on a wall in graduate school.  I do not know its original source.
  • Shortly before finishing this blog entry, while searching around for the source of the equation, I noticed that Alexander Bogomolny has written a closely related and quite detailed page at Cut-the-Knot.org. (Naturally, the level of mathematical sophistication of that page is much higher than this one.)