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 Subject: RE: exponents Author: Mathman Date: Oct 18 2004
On Oct 18 2004, Susan wrote:
> Even after teaching the students why x^-1 is 1/x in the same fashion
> as Mathman does, two months later, even after completley
> understanding the derivation, students will write 2^-1 = -2.

For what it's worth:

Some get confused over much less than that, and in much shorter a period of
time.  Perhaps we might reflect on the number who do not get confused, and
compare numbers to see whether or not any partiular approach is to be considered
effective.  Also, we might reflect upon our own education, and note our own
success or failure to understand then and now.

Consider my argument again:

Empirically [if we do the calculations], it is found that 10^5 / 10^3 = 10^2.
Several other examples, such as 5^7 / 5^4 = 5^3, lead to the inevitable
conclusion that in general we subtract exponents when dividing.  That is all
that needs to be remembered, as well as the fact that in each case, the exponent
is a convenient notation for common numbers.

So we carry that argument for any numbers, and find again empirically, as well
as from the general rule already established [that is, in the same way] that a^x
/ a^y = a^(x-y).  It then follows that this same exponent notation works for
other values without loss of generality. So a^(x-y) can be negative, and the
question is then what does this mean.  Also, that value was obtained from an
expression like 5^4/5^7. That is, using the rule already established, we find
that...

5^7/5^4 = 78125/625 = 125

We also note that 5^4/5^7 = 625/78125 = 1/125 making them reciprocal to each
other.

We note again that 5^7/5^4=125=5^3, and that 5^4/5^7=5^-3 from the rule for
division for exponents, and this is the same as 1/125.

Once a rule is established, it needs only to be memorised and put to further
use.  Forgetfulness by the student does not diminish its correctness or
usefulness.  If there is a better, or equally efficient way to do this study I
would be grateful to know of it and pass it along.  As they say ...whatever
works, and we really do learn something new every day.  Students will often make
up their own rules such as, "If x occurs twice on the same page, cancel."  We've
usually seen it all by the time we retire.

With regard to memorisation:  Some "tricks" and devices like SOHCAHTOA to
memorise the trig functions, or FOIL for binary multiplication  are useful, but
the real learning occurs when the student applies the function to definition for
angles and solution of triangles in any position with much practice.  It is the
practice and use that embed the ideas more than any other device, and young
people today get neither sufficient sample, nor variety, having to spend more
time with other associated studies in the same subject.

David.