How are they different? Cube root vs the exponent of 1/3

References
Complex Variables and Applications, James Ward Brown, Ruel V. Churchill, 7th edition, Chapter 1 Section 8, Roots of Complex Numbers

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[christopherkopperman]

The take away is that math is a language, and like all languages, context matters. So many forget that math is just a way we are trying to describe a concept.

[j.u..n]

When Graham Bell invented the mobile phone, he had 2 missed calls from the director of Wolfram.

[MattNolanCustom]

This (roots equispaced around a circular locus in the complex plane) was literally the *one* single moment in A-level maths lessons where I perked up and thought "ooh, that's interesting". Of course, the maths teacher immediately followed that with "but we don't go into that on this course".

[schizophrenicenthusiast]

"This angle theta will be theta" Great wording right there.

[mike.]

This really is dependent on whether you're talking about the real or complex power function here. Even then we sometimes fudge this line since we'll define square roots of negative numbers to be imaginary numbers. (-1)^(1/3) is indeed -1 if interpreted as the real power function, but if interpreted as the complex power function, the definition I use most defines that as the set of all complex cube roots of -1 rather than just the principal root.

[Math_Rap_and_GOP_Politics]

The Principal Root equation is taught at your University's graduate level Complex Analysis class. However, the fractional exponent notation is a concept more often utilized in an Abstract Algebra Course where your Commutative Ring must be specified. Your choice of Commutative Ring is the Complex Field. However, in some settings, say the Ring of Integers Modulo 5, 3^(1/3)=2 since 2 is that element whose cube is 5 in this setting.

[jimmonroe]

They are equivalent statements. WolframAlpha provided the real root for the cube root and principal root for the rational exponent. The real roots are the same for both expressions, and the principal roots are the same for both expressions.

[theprof]

So basically, they ARE the same. You are just picking a different principal value.

[EugeneKhutoryansky]

This is really a question of social convention, rather than a principle of mathematics. Aliens would have the same mathematics, but they would also have completely different definitions for all these terms.

[Robbedem]

We learned it slightly different.
For us, both are the same.
But you always have to specify when you are going to use complex solutions or not.
So it seems that (from your video) some cultures give one version the implied complex version.
I can see that being handy, but also causing confusion for those (like me) that don't follow the same convention.

[TranquilSeaOfMath]

7:34 This is important; thank you for your thoroughness. People need to be aware of this.

[hotflashfoto]

I didn't understand much of it at all, but I trust that you know what you're talking about from having watched a boatload of your videos and having been a subscriber for a long time.

[milanfanas]

I don't remember having studied this (but time has passed), it is very interesting indeed and makes sense

[goupil]

Isn’t the n-th root of x defined as x^(1/n)? The difference in Wolfram being standards of the principal value from it, like the first point rotating from 1 anti-clockwise.

[petersievert]

In lower classes we actually do teach at school that √x is only to be used for non-negative x and open up the definition later in A-level classes. It really depends a lot, so I dare say ³√-1 can also be "not defined" for good reasons.

[corvididaecorax]

So in other words both answers are correct answers to both equations, and it really just depends on what sort of answer you want. Sometimes you want all the possible answers, like often when solving the quadratic formula.

[StuartSimon]

I believe it is up to the individual product to treat the two root notation and the power notation the same or differently. To state that an odd root of a negative real number and its reciprocal power are treated as the real and principal such roots, respectively, seems to me to just happen to be how Wolfram does it. They could have just as easily defined the opposite.

[bb]

Still not clear to me. By definition these terms are identical. But you are saying they’re not. Why?

[mathmandrsam]

You can treat them differently and yes, their expressions lend themselves to different approaches to resolve/solve/simplify. But mathematically they are the same.

[Pengochan]

One really important part (briefly mentioned) is, that the convention is one thing, but often what matters is the context for a particular problem. The other thing is, that conventions may differ, so sometimes (e.g. when writing a paper) the question arises, how to express something unambiguously without much hassle.