|Re: fractional powers of negative numbers|
Message #3 Posted by Chuck on 20 Nov 2007, 6:58 p.m.,
in response to message #1 by Edward McNally
This is a great calculus question, because many calculators and software programs don't always show what is expected.
The reason it doesn't show the negative portion is because there are actually three cube roots of a number. Using DeMoivre's formula for roots, the first root (smallest angle in polar form) of a positive number is the positive root, so its plotted just fine. The first root of a negative number is complex, which is not plottable in the real plane. "Smart" calculators tend to give the complex root and thus give nothing for negative input. "Dumb" calculators stick with the "odd root of a negative is a neagive" idea, and plot the desired negative region. I'm pretty sure the HP50, TI89(?), Mathematica, all do not give the negative side. However, the "dumbed-down" TI84 gives the entire graph.
Some work arounds...
>> use the nth root button instead of x^(1/n)
>> rewrite x^(4/3) as (x^4)^(1/3)
>> on the HP35s write a negative number as -5i0, and then ^(1/n)
Good luck. But remember, you always need to be smarter than your calculator. It similar to the question of finding the all the solutions to the equation x^6-e^(.0001x) = 0 (or graphing the function on the left). Don't trust your calculator.
Edited: 20 Nov 2007, 8:58 p.m.