In preparing for a calculus exam, the text showed a graph of the function x^4/3  4x^1/3. The calculator showed no data for x<0 although the text had a graph extending from 4<x<4. I tried calculating some simple examples, such as (8)^1/3 and
(27)^1/3, which should be 2 and 3, respectively. The calculator returns (1.000, 1.732) and (1.500,2.598). These answers ,if squared, added and squarerooted, add up to 2 and 3 respectively. But I have no clue as to why I am getting these seemingly arbitrary "vector components". Any help?
fractional powers of negative numbers


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11202007, 03:50 PM
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11202007, 05:57 PM
Edward, This has to do with how you treat integers vs real numbers in exponential functions. For a^1/n, Your calculator is using 1/n as a real number, not the nth root of a. Hans
11202007, 06:58 PM
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 "dumbeddown" TI84 gives the entire graph.
Some work arounds... >> use the nth root button instead of x^(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^6e^(.0001x) = 0 (or graphing the function on the left). Don't trust your calculator. Cheers.
Edited: 20 Nov 2007, 8:58 p.m. ▼
11212007, 12:38 AM
Thank you. I do realize the calculator must sometimes make a choice between various solutions. I will use your thoughts for further study, and to further understand my calculator  which is an almost unbelievable machine. Thanks again.
11212007, 02:39 AM
Hello Gentlemen, ▼
11212007, 02:46 AM
Quote:My appologies, I meant to say the complex number [.99999, 1.732] :') Hal
11212007, 03:41 AM
Quote: Neither my 50G nor my 48G plot x^(1/3) in quadrant III. I suppose we have different flags set or otherwise different setups. Could you enlightend me on how to do it? Thanks in advance. I can however plot y= 3rd root of x in quadrants I and III on both the 48G and the 50G.
Edited: 21 Nov 2007, 4:31 a.m.
11212007, 04:24 AM
Hi Hal.
Quote: On my 50G, solving for (8)^(1/3) in exact mode gives me 2*e^(i*pi/3)... Best regards. Giancarlo
11222007, 04:29 AM
Thanks for your help. First, using the root function helped.
11272007, 08:45 AM
I had to check this problem with Derive for Dos (running on an Poqet PC Plus, btw a good replacement for a TI 92 if you allow non RPN every now and then): (8)^(3/4) simplifies to 1 + sqrt(3) ibut solve(x^3=8,x) will give you ie. you can't plot x^(1/3) for negative values straight forward. Interestingly, (8)^(4/3) is simplified to 88 sqrt(3)i, but no solution for x^(3/4)=8 is found. I wish there would be calculator with complete implementation of complex numbers, and a way to specify the domains for the calculation. Till then you still need to use your brain...
11212007, 01:27 AM
Quote:
They are not "vector components", but complex numbers with real and imaginary part that should read 1+1.732*i and 1.5+2.598*i  I'm sure you knew that. Edited: 21 Nov 2007, 1:59 a.m.
11212007, 02:31 AM
Edward  The sodescribed "smart" calculators are returning the primary roots, whose polarcoordinate angles  as measured counterclockwise from the positive real axis of the complex plane  are the smallest. The primary root of a negative number is complexvalued. Certainly, the negative realvalued root is also of interest if the range of the function is strictly realvalued. Calculators such as the HP35s, HP33s, and HP33SII can also return the real cube roots of negative numbers, using the "xth root of y" or cuberoot functions.  KS
Edited: 21 Nov 2007, 3:21 a.m. ▼
11212007, 04:26 AM
Quote: I've always seen it called the "principal" root. http://mathworld.wolfram.com/CubeRoot.html
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11252007, 04:24 AM
Hi, Rodger  Oops! A "misremembering" of accepted terminology on my part. "Principal" does make more sense than "primary", when one considers it.
"Primary" = "first stage", as in primary education, primary election, or primary sewage treatment. Of course, the principal root is also the first, or primary, root in the order of identification (lowest positive angle in the complex plane). Less than a year ago, I got the term right:  KS
11212007, 04:37 AM
There's an easy way to see all the nth roots of a negative number on the HP48G and descendants such as the HP50G. Let's say you want the nth roots of i. Just solve the equation x^n + i = 0. To do this you use the PROOT (polynomial solver) function. Remember to insert zeroes for the missing powers of x. Say you want the 7th roots of 3, of which there are 7. You need to solve; x^7 + 0^6 + 0^5 + 0^4 + 0^3 + 0^2 + 0^1 + 3 = 0 The "0" character is a zero. PROOT expects the coefficients of the polynomial in a vector, so type in [ 1 0 0 0 0 0 0 3 ] and then execute PROOT. To find the 3 cube roots of 8, type [1 0 0 8] PROOT.
11212007, 05:42 AM
Quote:You can plot this on the HP 50G (same on the 48G with slightly different syntax). It seems to be a question of operator precedence. You have to key in y= (3rd root of x)^4  (3rd root of x)*4
That will result in a graph on both sides of the y axis. Edited: 21 Nov 2007, 5:47 a.m. 