As you can see,I want to find the real-valued root in Mathematica, what should i do? I tried so hard to search it up online, but in vein.
Maybe try Surd instead of Sqrt.
Where do you mean I should try Surd with?
I mean I use 1/3 power, isn't that the same?
Another way, using CubeRoot
CubeRoot[7 + Sqrt] + CubeRoot[7 - Sqrt] // N
Thanks, it works, but is that really the only way to solve it? What if there are something like 1/5 power and the answer still come up with imaginary number?
Then you can use Surd, as Henrik suggested
Surd[7 + Sqrt, 5] + Surd[7 - Sqrt, 5] // N
CubeRoot is just a special form of Surd. The following will give the same result as if CubeRoot was used, in case of power 1/3:
Surd[7 + Sqrt, 3] + Surd[7 - Sqrt, 3] // N
The 1/5 power in Mathematica is defined as x^(1/5) == Exp[(1/5) Log[x]]. When x is negative, the principal value of the logarithm has I Pi as imaginary part, so that
x^(1/5) == Exp[(1/5) Log[x]]
In:= (-1)^(1/5) == Exp[(1/5) Log[-1]]
In:= Exp[(1/5) Log[-1]] // N
Out= 0.809017 + 0.587785 I
This issue is an endless source of confusion and frustration for beginners, which had been raised in a real-centered theory of roots.
Thanks everyone for helping.