Infinite Sequence

Problem

If $x^{x^{x^{x^{\dots}}}} = 2$, what is $x$?

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Hints

Hint 1

Think about defining the infinite power tower as a variable.

Hint 2

Let $y = x^{x^{x^{\dots}}}$ and write an equation in terms of $y$.

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Solution

Show Solution

$\sqrt{2}$

Show Explanation

Let’s carefully break this down. The expression is:

$$x^{x^{x^{x^{\dots}}}}$$

which means an infinite stack of exponentials with $x$ repeating over and over. Because it goes on forever, let’s name the entire tower $y$. So we write:

$$y = x^{x^{x^{x^{\dots}}}}$$

But notice that the part in the exponent (the tower after the first $x$) is exactly the same tower again — still $y$. Therefore:

$$y = x^y$$

From the problem, we know the whole tower equals 2:

$$y = 2$$

So substitute:

$$2 = x^2$$

Solving for $x$ gives:

$$x = \sqrt{2}$$

So the value of $x$ must be $\sqrt{2}$.