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This is kind of a spinoff on my question Divide by a number without dividing.

Can anyone think of some clever ways to raise any given number to any given power without using an exponent anywhere in your equation/formula?


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Anti-log of $y\log x$. – Gerry Myerson May 16 '13 at 13:04
Your title says $n$th power, which implies $n$ is an integer, but your question says $x^y$ where $y$, by implication, is not an integer. Integer powers can be efficiently compute using the Exponentiation by Squaring method. – Thomas Andrews May 16 '13 at 13:10
@AlbertRenshaw, "exponentiation by squaring" involves no exponentiantion, just multiplications. – vonbrand May 16 '13 at 13:15
There's only one base in grownup mathematics, and that's $e$. Anti-log of $Q$ is a way of writing $e^Q$ without writing an exponent. – Gerry Myerson May 16 '13 at 13:15
@GerryMyerson, does information theory not count as grownup mathematics? It uses base 2 more than $e$. – Peter Taylor May 16 '13 at 14:14
up vote 3 down vote accepted

You can always use the Taylor series for $f(u) = e^u$.

$$ x^y = 1 + y \ln x + \frac{(y \ln x)(y \ln x)}{2!} + \frac{(y \ln x)(y \ln x)(y \ln x)}{3!} + \cdots $$

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Very nice! :) Factorials have always fascinated me, it seems like they would be an ideal way to exponentiate since they involve sets of multiplication. – Albert Renshaw May 16 '13 at 14:11
Now that I understand more intuitively what logarithms actually are I redact my above comment. This is still a nice solution though :D – Albert Renshaw Oct 26 '15 at 20:29

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