Is $a^n$ an exponent of $a$? I see that definition says that $n$ is an exponent. But, the name of the function is normally expanded to the results computed by the function. That is, if we raise $a$ into power $n$ we say that result is a power (or exponent) of $a$. For instance, you say a $\sin\pi$. You apply the term 'sine' to the value of the function. So, here both $n$ and $a^n$ must be exponents. Right? 
Identically, WP says that power is the same as exponent in the context of exponentiation. Are both n and $b^n$ powers of b?
 A: It might be useful to list a number of related bits of terminology:
$a^b$ is a power of $a$.  In this expression, $b$ is the exponent, and $a$ is the base.  
If we think of $a$ as fixed and $b$ as varying, then the function sending $b$ to $a^b$ is an exponential function (with base $a$).
If we think of both $a$ and $b$ as varying, the function sending $a,b$ to $a^b$ is exponentiation.
A: Sure, you can say what you want. But it is difficult to communicate well with people who hold opposing conventions, and unfortunately, this is a case where pretty much every mathematician uses the same terminology (or would if they didn't make mistakes).
One thing that makes things difficult is that there are two distinct ways of looking at $b^p$. By definition $b^p$ is a number. However, it is useful to think of it as purely a symbolic expression, which would have meaning even if it didn't "evaluate".
Learning a new language is always difficult, and mathematical language is no exception. The easiest way to learn is to listen to others, read good works, and be entirely willing to change the way you use words when someone corrects you. That said, here's my best attempt at giving you "the rules"...


*

*$b$ is the base of the symbolic $b^p$.

*$p$ is the exponent of the symbolic $b^p$.

*the number $b^p$ is called the $p^\text{th}$ power of $b$.


When you are trying to emphasize different things about the symbolic expression, you call the expression different names:


*

*If $f(x)=b^x$, we say that $f$ is an exponential function.

*If $g(x)=x^p$, we say that $g$ is a power function (although, depending on the context it may be more appropriate to call it a polynomial or monomial function)


We speak $b^p$ by saying "$b$ to the $p^\text{th}$ power" or "$b$ to the $p$" for short, or some such thing.
These terminologies are identical when the base is a matrix. 
(Thinking of the matrix as an exponent has not been around for as long, so the terminology tends to be a bit different; one that comes to mind is that $b^P$ is usually called a matrix exponential or an exponential if the matrix part is obvious. But you still would not call $b^P$ an "exponent" or even a "matrix exponent", those would both be used to refer to $P$.)
In casual conversation, $p$ is sometimes called "the power", but this -always- means that $p$ is the power in the symbolic $b^p$, not the power of $b$ (because the power of $b$ is the number $b^p$. This is common in power functions, where $p$ is typically called "the power" (or even more questionably "the power of $g$") because a power is thought of as a number, and $x^p$ is not a number as long as $x$ is indeterminate.
