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?


  • $\begingroup$ Anti-log of $y\log x$. $\endgroup$ – Gerry Myerson May 16 '13 at 13:04
  • 1
    $\begingroup$ 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. en.wikipedia.org/wiki/Exponentiation_by_squaring $\endgroup$ – Thomas Andrews May 16 '13 at 13:10
  • 2
    $\begingroup$ @AlbertRenshaw, "exponentiation by squaring" involves no exponentiantion, just multiplications. $\endgroup$ – vonbrand May 16 '13 at 13:15
  • 1
    $\begingroup$ 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. $\endgroup$ – Gerry Myerson May 16 '13 at 13:15
  • 4
    $\begingroup$ @GerryMyerson, does information theory not count as grownup mathematics? It uses base 2 more than $e$. $\endgroup$ – Peter Taylor May 16 '13 at 14:14

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

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

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.