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cubic root of x vs. x power 1/3
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f(x)=x^(1/3) ==> x to power of 1/3
g(x)=nroot(x,3) ==> cubic root of x.
The question is, AFAIK and what my Dr. in Maths explains to me that both functions f and g has different definition area. f(x) is defined for x>=0 while g(x) is defined for all x.
However, Geogebra draws both the same, defined for all x (See attachment)
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Yes, they are the same in GeoGebra. Try x^0.333333333333333333
It's not that clearly set in stone.
Usually in math we say that a noninteger exponent is only allowed for a nonnegative base. That makes sense, because for almost all noninteger exponents this couldn't be defined meaningfully anyway, at least not within the real numbers.
That said, one could make an argument that for rational exponents with an odd denominator the power can be defined in a meaningful way even if the base is negative, because negative numbers have odd roots that are real numbers. And it seems that Geogebra chose to go this route. Generally, one should always be wary when dealing with such powers with a negative base, especially when using power laws (they don't always apply then).
Btw, some math books also only define the root operators (square root, cubic root, etc.) only for nonnegative numbers, but some allow this for the odd roots. The reasoning is the same as for the powers: it kind of makes sense. But it's also somewhat problematic when operating with those roots, because the root laws don't always apply.
I guess allowing negative numbers feels a bit more comfortable for roots than for powers, because roots are more restricted from the start (they are basically powers where the exponents are restricted to unit fractions), and that might be the reason why math books might forbid this for powers but still allow this for the odd roots. Which seems to be the stance of your math teacher or the books he works with.
My rule of thumb is: If you know what you are doing (in particular what you can and what you can't do when negative numbers are involved), feel free to define odd roots or powers with certain rational exponents also for negative numbers and work with them as you see fit. Geogebra does it, and I am assuming that Geogebra takes care to handle those properly. (That said, I am still extra careful when using such things with Geogebra (or any other tool) and don't trust any result blindly.)
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