Power or Exponent?

Date: 10/14/2003 at 10:35:28
From: Beth  
Subject: Exponents

Our 5th grade textbook (McGraw Hill) defines "power" as "a number 
obtained by raising a base to an exponent".  I have never seen this
word defined in this way.  

I have always used the words "power" and "exponent" interchangeably. 
For example, 2 raised to the 3rd power, or 10 to the 6th power.  Is
the textbook's definition correct?

I don't see how you can raise something to a power, and get the power
as the answer. 

Date: 10/14/2003 at 12:54:12
From: Doctor Peterson
Subject: Re: Exponents

Hi, Beth.

The book's terminology is correct; but the English usage here is 
awkward and is often misinterpreted, not only by students but even by 
textbook authors and lexicographers.  That, of course, means that you 
can find many authorities for a different view than mine!  In fact, my 
understanding of the terms may be a minority view; but I think it is 
correct.

When we write

   3
  2

we say that 2 is the "base", 3 is the "exponent", and the whole thing 
is "a power of 2" (in particular, "the third power of 2").  The 
awkwardness comes from the fact that we call this expression "2 to the 
third power" or "2 raised to the power [of] 3".  This SOUNDS as if we 
were saying that 3 was the "power" to which we raised 2, and as a
result the word "power" is, as you point out, often used 
interchangeably with "exponent".

But if you look closely at the phrase "raised to the third power", you
see that we are not saying that 3 IS the power; rather, 3 IDENTIFIES
which power you are talking about, the third one.  We are raising the
number 2 TO a power, changing it from its original "weak" form to a
more "powerful" form; in fact, we have raised it to its third power,
the third level it can reach.  The power is the number it got to, not
the number of steps it took to get there.

I believe that the phrase started with "the third power of 2", which 
clearly names the result of the operation as a power; then moved to "2
raised to the third power", which means the same thing but emphasizes
the operation of "raising" rather than the value; and then, when
variable exponents were needed, had to be twisted around to "2 raised
to the power of x" to avoid having to say "2 raised to the xth power".
And at that point, it started sounding as if x was a power, though
even here you can still see a distinction, in that the power is "of
x", that is, belonging to, or associated with, x, not x itself.  When
people say "2 raised to the power x", that distinction is lost.

So I can't blame people for getting confused, and I have to recognize 
that the term "power" is very commonly used to mean "exponent".  But I 
think it is useful to retain a word that refers to the whole 
expression (just as we use "product" to refer to the result of 
multiplication); and the only word available is "power"!  Can you
think of an alternative, if we reserve "power" to mean the exponent?

To support my theory, note that the first use of "power" in this sense
meant the result of the operation, not the exponent:

  Earliest Uses of Some Words of Mathematics 
    http://jeff560.tripod.com/p.html 

  POWER appears in English in 1570 in Sir Henry Billingsley's
  translation of Euclid's Elements: "The power of a line, is the
  square of the same line." 

Here multiple powers are not yet in view, so no counting of powers is
needed; the square is THE power.  But the square is the result of the
operation, clearly not the number 2.

Similarly, see

  Math Words, and Some Other Words, of Interest
    http://www.pballew.net/arithme8.html#power 

  Power: The word power comes from the French poeir and perhaps the
  earlier Latin word potere from which we get potent. Both words
  refer to ability or being able. In mathematics, power refers to
  the number arrived at by raising a number to an exponent. In the
  mathematical expression 3^2=9, three is the base, two is the
  exponent, and nine is the power. Students often refer to the
  exponent as the power, but this is not historically correct,
  although it has become so common, even among many teachers, that
  some dictionaries refer to the power as the exponent. 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/