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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