Thanks for this info about the brach cut discontinuity.

But I'm not exactly sure how this fits what the results of the two plots show. since Power[] is being used the same in both cases (the only difference is Power[] is nested once more in the first case than the second) it would seem both would have the "gap" (missing portion from -1 to 1).

But, how can this branch cut discontinuity account for the missing portion of the plot from -1 to 1, but yet it is happy from minus infinity to -1.

In other words how does this account for both the absence of the entire center portion (from -1 to 1) in the second Plot but the presence of the left portion (from minus infinity to -1) in the first Plot?

Also the portion of the plot from (0,1) is missing as well on the second and also the second shows the portion of the plot from minus infinity to -1.

It would seem that if one is missing then they they both should be. And what about the fact that part of the center portion is from (0, 1) which is not part of the discontinuity region?

Plus none of these even involve any complex values (since the root is computed after the exponential so none of the intermediate values will ever be negative when the root is computed, since they are raised to an even power beforehand).

Do you see my dilemma with this issue?

Thanks again...

I teach my students that raising a negative number to a noninteger power is very dangerous, because all algebraic rules stop working. For example,

x^(n/m)==(x^n)^(1/m)==(x^(1/m))^n

is true if x>0, but it is trouble if x<0. I think Mathematica does the right thing in this instance. Try

N[(-2)^(4/5)]
N[((-2)^4)^(1/5)]
N[((-2)^(1/5))^4]