Solving Nested Exponents - A Programmer's Headache

[zyflair]

Hello everyone, I have stumbled across a curious question while programming.

To start the process, say I have a a positive integer greater than one: α₀
Nesting that in an exponent is a simple operation and a common occurrence: α₀^α₁
(α₁ is also a positive integer greater than 1. In fact, all numbers mentioned in this problem of mine are in that given set)

But then I run into a problem when I nest this number with yet another exponent: α₀^α₁α₂
See, the exponent for me is calculated from right to left, so I can't simply store α₀^α₁ into memory and then set that to the α₂ power, because that only works on operators that are calculated left to right.

So my question is this: Given a chain of nested exponents α₀^{α₁α₂...α_n}, is there a closed form means to determine α₀^{α₁α₂...α_nα_n+1} without having to recalculate the exponents from right to left?

I apologize if this is the wrong place to ask, but as a programmer, nested exponents are causing me a headache, because I have to spend n operations to calculate the nested exponent every time I nest it. While not completely slow, I would enjoy a faster means of computing this. I am also aware my question might not seem completely clear, so please ask for clarifications if you require any.

Thanks for your time.

 

[symbolipoint]

Try reviewing rules of exponents using some simple example.

Find $2^{{3}^{2}^{5}}$. Try doing the exponents from right to left; and also try doing from left to right. See what results.

format tags not working right. 2^3^2^5
Using TexAide: $\[<br /> 2^{3^{2^5 } } <br /> \]$

[STRIKE]Should be the same as $2^{3*2*5}$[/STRIKE]Obviuos mistake. That would not be result of the nesting. See next member's post.

 

[rcgldr]

Try reviewing rules of exponents using some simple example.

By nesting the OP means 2^(3^4) = 2^81 = 2417851639229258349412352, this is different than (2^3)^4 = 8^4 = 4096.