Exponents Issue

[dimchansky]

Expecting the expression

Code:

x^(1/9)*(x*x^(1/3))^(1/6)

to be simplified to

Code:

x^(1/3)

, but the calculator does not simplify it (tried collect, simplify functions with no effect). TI-89 Titanium and TI-Nspire both calculators simplify it.
Any ideas?

[CyberAngel]

(02-04-2020 11:23 AM)dimchansky Wrote:  Expecting the expression

Code:

x^(1/9)*(x*x^(1/3))^(1/6)

to be simplified to

Code:

x^(1/3)

, but the calculator does not simplify it (tried collect, simplify functions with no effect). TI-89 Titanium and TI-Nspire both calculators simplify it.
Any ideas?

A funny try in the middle of the night:

Code:

#cas
expcollect(f):=
BEGIN
  LOCAL g,s,v;
  v:=mat2list(algvar(f));
  s:=SIZE(v);
  v:=head(mat2list(v(s)));
  g:="'solve(y_="+f+","+v+")'";
  g:=expr(g);
  f:="'solve("+v+"="+g+",y_)'";
  RETURN expr(f);
END;
#end

VPN

[Claudio L.]

(02-04-2020 11:23 AM)dimchansky Wrote:  Expecting the expression

Code:

x^(1/9)*(x*x^(1/3))^(1/6)

to be simplified to

Code:

x^(1/3)

, but the calculator does not simplify it (tried collect, simplify functions with no effect). TI-89 Titanium and TI-Nspire both calculators simplify it.
Any ideas?

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.

[CyberAngel]

(02-05-2020 10:42 PM)Claudio L. Wrote:  

(02-04-2020 11:23 AM)dimchansky Wrote:  Expecting the expression

Code:

x^(1/9)*(x*x^(1/3))^(1/6)

to be simplified to

Code:

x^(1/3)

, but the calculator does not simplify it (tried collect, simplify functions with no effect). TI-89 Titanium and TI-Nspire both calculators simplify it.
Any ideas?

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.

Right!
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 TI-like simplification.

[Albert Chan]

(02-05-2020 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:

• (a*b)^c → a^c * b^c
• (a^b)^c → a ^ (b*c)
  

To force above rules for other c, use PowerExpand, but result might not be correct.

x^(1/9)*(x*x^(1/3))^(1/6) // PowerExpand    → x^(1/3)

(02-05-2020 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)).

[CyberAngel]

(02-06-2020 12:21 AM)Albert Chan Wrote:  X

(02-05-2020 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)).

Try 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

[swagner53]

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.

[dimchansky]

(02-06-2020 09:19 PM)swagner53 Wrote:  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.

That's interesting.. Because I get the different result on the same version, but on Emu.

[dimchansky]

(02-05-2020 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:

[DrD]

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 manually-derived result.

[roadrunner]

Perhaps this is helps:

   

-road

[dimchansky]

(02-07-2020 02:21 PM)roadrunner Wrote:  Perhaps this is helps:

But how did you get the first simplification result? Doesn't work for me:



What's the version of your calculator? Hardware/Emu?

[roadrunner]

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

[dimchansky]

(02-07-2020 04:26 PM)roadrunner Wrote:  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.

doesn't work on emulator, it returns the same answer after simplification.

[roadrunner]

Here's something interesting:

Copy your expressing into the emulator and click enter
Clicking the expressing on the right and clicking simplify gives a different answer than clicking the expression on the left?!??!?!?!

   

-road

[roadrunner]

Here's a better example:

   

-road

[thenozone]

wolfram alpha give me:- see attachments,
but i could have mistyped.

[swagner53]

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.

[dimchansky]

(02-07-2020 04:41 PM)roadrunner Wrote:  Here's something interesting:

Copy your expressing into the emulator and click enter
Clicking the expressing on the right and clicking simplify gives a different answer than clicking the expression on the left?!??!?!?!

yes, this way it works! it's some kind of black magic!

[Claudio L.]

(02-07-2020 08:41 PM)thenozone Wrote:  wolfram alpha give me:- see attachments,
but i could have mistyped.

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 = (-8-8*√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.