2

Or would "powers of $x$" in those examples refer to $y$ and $z$, respectively?

JustAsking
  • 307
  • 3
    The English term for $y$ or $z$ would be "exponent". –  Jan 03 '21 at 20:33
  • @Gae.S., that was the answer I suspected and, frankly, was hoping for. I am relieved I will not have to fight terminology in order to succinctly state certain results in my writing. – JustAsking Jan 03 '21 at 20:38
  • 2
    I think "power" works too - reminds me of "power series", and makes me wonder why we don't say "exponent series"... –  Jan 03 '21 at 20:59
  • 2
    I think it's fairly colloquial and I'm not sure there is an official definition. One will say "$x$ to the $k$ power" with implies that $k$ is "the power" but I have heard people say, and have said "$64$ is a power of $2$" so mean $64=2^k$ for some integer $k$. Now that I think about it those concepts are irreconcilable but they are informal. I THINK I stick with $x^y$ is "exponential expression"; $x$ is the base; $y$ is the exponent and if the exponent is an integer we can call it a "power". I don't know though. It's fairly contextually driven. – fleablood Jan 03 '21 at 21:00
  • 2
    @WilliamBarnes Possibly a "power" implies an integer exponent. After all there is nothing particularly interesting about $x^{\log_x w} = w$ and $x$ and $w$ are arbitrary (not $1$) positive reals. – fleablood Jan 03 '21 at 21:03
  • 1
    @WilliamBarnes Huh? We're not summing the exponents! – Ted Shifrin Jan 04 '21 at 02:19

1 Answers1

1

In the expression "$x^y$", $y$ is the "exponent"/"power" and $x$ is the "base".

But if $x$ is an arbitrary real number, I would not generally call $x^y$ "a power of $x$", though I might call it "$x$ raised to a power".

"Powers of $n$" where $n$ is a particular positive integer (or rarely, perhaps a positive rational) refers to the values $n^m$ where $m$ is an integer. In an integer-only context, the "powers of $n$" might require $m$ to be nonnegative or even positive. But there are also contexts where we allow $m$ to be a negative integer (the famous film(s) Powers of Ten is one example). Curiously, a phrase like "negative powers of $n$" is common, because context makes it clear that it must be the exponent $m$ which is negative.

It's difficult to provide a definitive source for the meaning of this phrase because it is not often carefully defined in textbooks. The English Wikipedia page for "Power of two" has some references, though.

Mark S.
  • 23,925