Understanding quadratic root

Question

Quadratic root is defined as $\sqrt{ x^2} = |x|$. Easy to remember, but seems to lack logic. And this topic is about you proving me wrong.

1) This definition of a square root is not universal and is restricted to one special case when $x \in R$.

2) And how do we define roots of higher orders then? $\sqrt[4]16 = 2,2i$ (I do not add two more negative answers because of that "absolute value" trick). That means $\sqrt[4]{x^4} = |x|$ is not a function as it returns two values.

3) "Square root has to return only one value, otherwise it is not function. And we want really badly square root to be a function, thus we force it to return an absolute value". Sounds pretty much like the most convincing explanation to me for such a definition. As of my understanding square root (just like roots of other orders) returns more than one value thus is not a function and should not be forced to resemble one

Answer 1

There are several functions here. As far as I remember:

1)
$\sqrt[2n]x$ for x real is defined for $x \geq 0$ only ($n\geq1$ is an integer).
It's a function returning a single real value which is non-negative.

2)
$\sqrt[2n+1]x$ for x real is defined for all real x ($n \geq 1$ integer).
It's a function returning a single real value (it can be negative).

3) $\sqrt[n]z$ for z complex is defined for all complex z.
It's a multi-valued function which returns $n$ complex numbers.
Note that if you take a real number and use that third definition,
it works OK. One of the roots returned here coincides with the root
/returned by 1) or 2)/, if such a root exists / as per 1) or 2) /.
So this third definition is more general than 1) and 2), it builds upon them.

Of course this idea of root can be further extended and refined,
but at a certain level of knowledge I think this idea of roots is OK
and makes total sense.

Answer 2

In the complex numbers, there are essentially two choices for defining $\sqrt[n]{z}$:

1. Make this a "multivalued function" with $n$ different values (for $z \ne 0$)
2. Choose a particular branch, i.e. for each $z$ one of the $n$ roots is singled out as the one you want. This can't be done in a way that is continuous on the whole complex plane: you will always have a "branch cut", a curve extending from $0$ to complex $\infty$ along which the function is discontinuous.

For example, the principal branch can be defined as follows: if $z = r e^{i\theta}$ with $r \ge 0$ and $-\pi < \theta \le \pi$, then the principal branch of $\sqrt[n]{z}$ is $ r^{1/n} e^{i\theta/n}$.