Exponents Issue

[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.