PRODUCT:  TI-81 AND TI-85
SUBJ: Graphing x^y

This note is posted to clarify the handling of x^y, especially
where y is a rational fraction, in response to a recent question
by Robert Garfunkel of Montclair State College.  His question
was why the graphs of x^(2/3) and (x^(1/3))^2 were not the same
for x<0.  Similar questions have been asked from time to time
and the answer may be of general interest.

To begin, let's establish some basic facts to make the discussion
clear:	(If you follow along with your calculator, remember that
the TI-81 and TI-85 negation key is lower precedence than ^, so
don't leave off the parenthesis in (-3)^.5, for instance.)

   If we have x^(1/q) where x is a real number and q is a positive
   integer, there will be q answers (roots).  These multiple
   values are called branches and the "principal branch" has
   magnitude (abs(x))^(1/q) and an angle that is (angle of x)/q.
   (You can get the angle of a complex number on the TI-85 with the
   angle function.  For the function to work right, remember to
   enter the number as a complex data type, ie (-3,0).	For
   positive real numbers the angle is zero, and for negative real
   numbers the angle is PI.)  As a consequence, when x is positive,
   the principal branch is always a real number. (Zero divided by
   q is zero.)	When x is negative, the principal branch is always
   complex for q>1.  (PI divided by a q>1 is never zero or PI.)
   However, for x<0 and q an odd integer, one of the branches is
   always real. (The branches for q=3 have angles PI/3, 3*PI/3 and
   5*PI/3.  The first branch is principal, the second branch is
   real.  The branches for q=4 are PI/4, 3*PI/4, 5*PI/4, 7*PI/4.
   When q is even, there is not a real branch.)

   If we have x^(p/q) where x is a real number and p and q are
   positive integers, and compute the result as (x^(1/q))^p, the
   result is real if x^(1/q) is real.  So, in general, x^y (for
   real x and y) has a real answer only for x>=0 or when y can
   be expressed as a rational number p/q, with q odd.  However,
   as mentioned above, the principal branch of x^y is not real
   for x<0 and y<1.

Now to digress for a bit of history.  The universal power function
is calculated as:  x^y = exp(y ln x).	The early scientific
calculators with a universal power or root function give an error
for x^y if x<0 (even (-2)^2, for example).  This is because
ln(-2) is complex and the early calculators did not have the memory
space to compute complex intermediate results or make special tests
to reformulate the problem.  However, for our more modern scientific
calculators, it is our standard practice to make appropriate tests
to return  4 for (-2)^2 and, since a root key is featured, to return
answers when a real root exists (for instance: cube root of (-8) or
(-8)^(1/3) returns -2).  It is this feature that seems to cause some
confusion when implemented on our graphing calculators.

The TI-81 and TI-85 (as well as the TI-68) return the principal
branch of x^y except when y can be expressed as a rational fraction
(1/q) and q is odd.  For this exception, the real branch is returned.
The capability to return real results for integer roots of negative
numbers can lead to some confusion, but we have found it to have
practical utility.

Specifically to the question that was raised by Mr. Garfunkel, the
principal branch of x^(1/3) and x^(2/3) are not real for x<0, but the
TI-81 and TI-85 plot the real branch for x^(1/3) due to the "odd root"
feature.  So (x^(1/3))^2 plots as a real branch, because an integer
power of a real number is always real.	The formulation (x^(1/q))^p
or (x^p)^(1/q) plots the real branch of x^(p/q) if a real branch
exists.

In the case of derivatives, if the numeric derivative function is
used (NDeriv on TI-81, nDer on TI-85), you get a graph of the
derivative over the same range that the function plots.  However,
the der1 and der2 functions on the TI-85 are exact derivatives
and are computed in a manner equivalent to forming the symbolic
derivative.  So, der1(x^(1/3),x) only plots for x>0 because the
derivative is 1/(3x^(2/3)), but der1(x^(2/3),x) plots for all x
(except x=0) even though x^(2/3) doesn't plot for x<0, because
the derivative is 2/(3x^(1/3)).

We hope this discussion helps you.  We welcome your comments
for future products.  Therefore, if you have an opinion or
preference concerning other possible ways to handle this function
please let us know through GRAPH-TI or by email to the address below.

1. Always return the principal branch for x^y.
2. Always return the real branch for x^y (the calculator would
   determine if y is equivalent to some (p/q) with q odd and
   return (x^(1/q))^p).
3. Provide a mode selection to allow the user to select either real
   or principal branches for multivalued functions.

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  Texas Instruments (Consumer Products)
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