How do we accurately define a negative raised to a fractional power, as it can be written in different ways which is misleading Negative to fractional (and decimal) power can be misleading :
For example:
$(-2)^{2.1}$  may be expressed as:
$(-2)^{(21/10)}$
= $\sqrt[10]{(-2)^{21}}$
= $\sqrt[10]{\text{negative}}$
= $\text{a complex number}$
OR:
$(-2)^{(210/100)}$, which is same as above
$= \sqrt[100]{(-2)^{210}}$
$=\sqrt[100]{\text{positive}}$
$= \text{a real value}$
 A: note that we have $$x^{f(x)}=e^{f(x)\ln(x)}$$ is only defined for $$x>0$$
A: The raising of a negative real (incl. integer) number to a rational  exponent has a sense only in the complex field, as
already commented.
Now, the complex power is defined to be a multivalued function in general, such as the familiar $z^{1/2}= \pm \sqrt{\,z\,}$.
Precisely it is defined as
$$
\begin{gathered}
  S = z^{\,w} \quad \left| \begin{gathered}
  \;S,z,w \in \;\mathbb{C}\; \hfill \\
  \;w = u + i\,v \hfill \\
  \;k\; \in \;\mathbb{Z}\, \hfill \\ 
\end{gathered}  \right.\quad  = \exp \left( {w\,\text{Ln}\,\left( z \right)} \right) = \exp \left( {w\,\left( {\ln \left| z \right| + i\arg \left( z \right) + i\,2k\pi } \right)} \right) =  \hfill \\
   = \exp \left( {w\,\left( {\ln \left| z \right| + i\arg \left( z \right) + i\,2k\pi } \right)} \right) = \exp \left( {w\,\left( {\ln \left| z \right| + i\arg \left( z \right)} \right)} \right)\exp \left( {i\;w\,\,2k\pi } \right) = z^{\,w} 1^{\,w}  =  \hfill \\
   = \exp \left( {\,u\ln \left| z \right| - v\arg \left( z \right) - \,2vk\pi  + i\,\left( {u\arg \left( z \right) + v\ln \left| z \right| + \,u2k\pi } \right)} \right) =  \hfill \\
   = \left\{ {s_{\,k} } \right\} \hfill \\ 
\end{gathered} 
$$
where $\text{Ln}$ denotes the multivalued logarithm, and $\exp(x+iy)$ is the entire function $e^x(\cos{y}+i \sin{y})$.  
The set will contain only one value if $w=n \in \;\mathbb{Z}$, $q$ values if $w$ is rational $=p/q$ with $gcd(p,q)=1$,
and infinite values if $w$ is irrational or complex.
Any of the values in the set $S$ will be such that 
$$
s_{\,k} ^{\,1/w}  = z
$$
However it comes out that is not possible to define the exponentiation/power in such a way
that the law of addition and multiplication of the exponents be always preserved
$$
z^{\,w_{\,1}  + w_{\,2} } \mathop  \ne \limits^{\text{in}\,\text{general}} z^{\,w_{\,1} } z^{\,w_{\,2} } \quad z^{\,w_{\,1}  \cdot w_{\,2} } \mathop  \ne \limits^{\text{in}\,\text{general}} \left( {z^{\,w_{\,1} } } \right)^{\,w_{\,2} } 
$$
Refer for instance to this Wikipedia article
Taking a case even simpler than the one you proposed, we have for instance that
$$
\begin{gathered}
  \left( { - 1} \right)^{\,1/3}  = \left( { - 1} \right)^{\,2/6}  = \left\{ {\exp \left( {i\frac{{2k + 1}}
{3}\pi } \right)} \right\} = \left\{ {\exp \left( {i\frac{1}
{3}\pi } \right),\;\exp \left( {i\,\pi } \right),\;\exp \left( { - i\frac{1}
{3}\pi } \right)} \right\} \hfill \\
  \quad  \subset  \hfill \\
  \left( {\left( { - 1} \right)^{\,2} } \right)^{\,1/6}  = 1^{\,1/6}  = \left\{ {\exp \left( {i\frac{{2k}}
{6}\pi } \right)\;\left| {\;0 \leqslant k \leqslant 5} \right.} \right\} \hfill \\ 
\end{gathered} 
$$
