Exponents Issue

02042020, 11:23 AM
Post: #1




Exponents Issue
Expecting the expression
Code: x^(1/9)*(x*x^(1/3))^(1/6) Code: x^(1/3) Any ideas? 

02052020, 12:26 AM
Post: #2




RE: Exponents Issue
(02042020 11:23 AM)dimchansky Wrote: Expecting the expression A funny try in the middle of the night: Code: #cas VPN 

02052020, 10:42 PM
Post: #3




RE: Exponents Issue
(02042020 11:23 AM)dimchansky Wrote: Expecting the expression I don't think both expressions are equivalent with x<0? I don't know, it seems to me some of the multiple solutions might get lost in the shuffle and that's why the CAS refuses to simplify. Wolfram Alpha seems to think it's a bad idea to simplify this too. 

02052020, 11:05 PM
Post: #4




RE: Exponents Issue
(02052020 10:42 PM)Claudio L. Wrote:Right!(02042020 11:23 AM)dimchansky Wrote: Expecting the expression Fractional or rational exponents using Power function differ from the Root function. Note that negative real numbers can have odd roots. A third root of 8 is 2. He asked for the TIlike simplification. 

02062020, 12:21 AM
Post: #5




RE: Exponents Issue
(02052020 10:42 PM)Claudio L. Wrote: Wolfram Alpha seems to think it's a bad idea to simplify this too. Mathematica "^" only simplify if c is integer:
x^(1/9)*(x*x^(1/3))^(1/6) // PowerExpand → x^(1/3) (02052020 11:05 PM)CyberAngel Wrote: Fractional or rational exponents using Power function differ from the Root function. I tried XCas using root, but it made no difference. From XCas help, it says root(a,b) returns b^(1/a) (root(2,3)=sqrt(3)). 

02062020, 04:06 AM
Post: #6




RE: Exponents Issue
(02062020 12:21 AM)Albert Chan Wrote: XTry on an emulator and on a real calculator CAS a third root or cubic root of minus eight A] NTHROOT(3,8) (minus eight) ←the surrounding () are mandatory to power one 3rd B] (8)^(1/3) Check the CAS complex off/ use i off and principal off What do you get? – – VPN 

02062020, 09:19 PM
Post: #7




RE: Exponents Issue
Hi,
I entered the equation in algebraic mode exactly as you had it, pressed enter, and got x^(1/3). I am on hardware C, version 2.1.14425, CAS 1.5. There seems to be some improvements is the new version. 

02072020, 09:03 AM
Post: #8




RE: Exponents Issue
(02062020 09:19 PM)swagner53 Wrote: Hi, That's interesting.. Because I get the different result on the same version, but on Emu. 

02072020, 09:38 AM
Post: #9




RE: Exponents Issue
(02052020 10:42 PM)Claudio L. Wrote: I don't think both expressions are equivalent with x<0? I don't know, it seems to me some of the multiple solutions might get lost in the shuffle and that's why the CAS refuses to simplify. Wolfram Alpha seems to think it's a bad idea to simplify this too. Yeah, it can be the case. But then, assume function does not help: 

02072020, 12:47 PM
Post: #10




RE: Exponents Issue
wxmaxima:
(%i1) x^(1/9)*(x*x^(1/3))^(1/6); (%o1) x^(1/3) For students studying laws of exponents, (with x being strictly symbolic), this result confirms a manuallyderived result. 

02072020, 02:21 PM
Post: #11




RE: Exponents Issue  
02072020, 02:26 PM
Post: #12




RE: Exponents Issue  
02072020, 04:26 PM
Post: #13




RE: Exponents Issue
On the emulator:
1. copy the expression from your first post in this thread and past onto the command line, click enter; 2. click on the expression; 3. click simplify; 4. click on the answer; 5. repeat from step 3. I just now tried it on the handheld but keyed in the expression manually, it returned identical results. hardware info: road 

02072020, 04:29 PM
Post: #14




RE: Exponents Issue
(02072020 04:26 PM)roadrunner Wrote: On the emulator: doesn't work on emulator, it returns the same answer after simplification. 

02072020, 04:41 PM
Post: #15




RE: Exponents Issue  
02072020, 04:52 PM
Post: #16




RE: Exponents Issue  
02072020, 08:41 PM
Post: #17




RE: Exponents Issue
wolfram alpha give me: see attachments,
but i could have mistyped. 

02072020, 08:54 PM
Post: #18




RE: Exponents Issue
If I enter the equation by itself and press enter, then I get x^(1/3)  no simplify. If I start with simplify, I get the same results as Roadrunner (cycle 3times with simplify and finally get x^(1/3). Note I have set the Simplify setting in CAS to Maximum.


02072020, 10:15 PM
Post: #19




RE: Exponents Issue
(02072020 04:41 PM)roadrunner Wrote: Here's something interesting: yes, this way it works! it's some kind of black magic! 

02072020, 10:58 PM
Post: #20




RE: Exponents Issue
(02072020 08:41 PM)thenozone Wrote: wolfram alpha give me: see attachments, You got it right, except that's the "mobile version", if you look at the desktop result, you'll see above the plots another box that says "result" and it shows the expression as X^(1/9)*(x^(4/3))^(1/6). So it only collapsed the x*x^(1/3) into x^(4/3) Also notice on your own pictures, the result at the bottom reads "Alternate form assuming x>0:", so it's not quite the same. The problem is that when x<0, any fractional power will choose the principal root, but depending on the order of operators, it may choose a different root. For example, assume x=8 Basic algebra tells you: x*x^(1/3) = x^(4/3) = (x^(1/3))^4 = (x^4)^(1/3) The principal root of (8)^(1/3) = 2*e^(i*pi/3) = (1+√3*i) (x^(1/3))^4 = (1+√3*i)^4 = (88*√3*i) = 16*e^(2/3*pi*i) That's a valid solution. Now the other form: (x^4)^(1/3) = 4096^(1/3) = 16 which is also a completely valid solution. If we had a system that calculates all roots no matter what, we'd have all 3 roots of x^(1/3) at the end on all the different possible simplification paths (we'd have 9 different roots with x^(1/9) then 6 of them will collapse into the same result, ending with only 3 roots, precisely the roots of x^(1/3)). So the simplification is not wrong per se, but evaluation of the two expressions may return different roots, which can be very confusing. To make it worse, it may change the chosen root as you change the value of x, so the plot may appear to be discontinuous when the expression is actually not. 

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