Discussion:  Roundtable 
Topic:  exponents 
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Subject:  RE: exponents 
Author:  Mathman 
Date:  Oct 18 2004 
I'm not trying to be argumentative or picky, Alan, and do appreciate the
difficulties some of the children have, but does not the sequence I suggested
lead to the *consequence* that the negative exponent represents the inverse?
The only definition (subtraction of exponents)is empirical from the start, and
all one needs to do is to use it directly to all possibilities. Defining a rule
that one must subtract exponents when dividing as shown [which follows from
observing empirically 10^5/10^2 = =10^(52) = 10^3 and so on] it *follows*
that 10^3/10^5 = 10^(2). Since it is already established from past study
that 10^5/10^3 and 10^3/10^5 are reciprocals, then the fact is one that follows,
and is not defined in and of itself. There are in fact some exponential
representations that can not offer an alternate representation, but this one
follows directly from observation. It's the way I taught it successfully for
many years, writing down a sequence of decreasing values to show the pattern of
number and of exponent.
David.
> On Oct 18 2004, Mathman wrote:
> Is it
(the fact that powers with
> negative exponents
are the reciprocals of the ones with positive
> exponents)
> not simply a result of the same empirical rule
> a^5/a^3 =
> a^(53), and so it follows for completion that a^3/a^5
> = a^(35),
> and likewise a^n/a^n = a^0 ?
It is certainly the
> most natural definition (and probably the only reasonable one), but
> not the only possible one. My point was just that it is still a
> definition (which we choose) rather than a fact of the kind that we
> have no choice about (such as a theorem  which is an unavoidable
> consequence of what we have already chosen). And I worry that by not
> making the distinction we may leave some students feeling frustrated
> at their failure to see as obvious something that is not even
> strictly true(in the sense that they may be interpreting it).
 
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