Not Mathcad But Latex code:
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\section*{Rigorous Evaluation of $\left(-\frac{1}{8}\right)^{-2/3}$}
The expression involves a negative base raised to a rational power, which is multi-valued in the complex domain $\mathbb{C}$. We define the operation using the complex exponential and logarithm:
\[ f(z) = z^a = e^{a \log(z)} \]
\subsection*{1. Simplification of the Base}
First, we apply the negative exponent to find the reciprocal:
\[ \left( -\frac{1}{8} \right)^{-2/3} = \left[ \left( -\frac{1}{8} \right)^{-1} \right]^{2/3} = (-8)^{2/3} \]
\subsection*{2. Polar Representation of the Base}
The complex number $z = -8$ lies on the negative real axis. Its magnitude $r$ and argument $\theta$ are:
\[ |z| = 8, \quad \arg(z) = \pi + 2k\pi \quad \text{for } k \in \mathbb{Z} \]
Thus, in exponential form:
\[ -8 = 8e^{i(\pi + 2k\pi)} \]
\subsection*{3. Application of the Rational Exponent}
Substituting this into the power function:
\[ (-8)^{2/3} = \left( 8e^{i(\pi + 2k\pi)} \right)^{2/3} = 8^{2/3} \cdot e^{i\frac{2}{3}(\pi + 2k\pi)} \]
Since $8^{2/3} = (\sqrt[3]{8})^2 = 4$, the general solution is:
\[ w_k = 4e^{i\left( \frac{2\pi}{3} + \frac{4k\pi}{3} \right)} \]
\subsection*{4. The Three Distinct Roots}
For $n=3$ (the denominator of the exponent), there are exactly three unique values for $k = 0, 1, 2$:
\begin{itemize}
\item \textbf{Case $k=0$ (Principal Value):}
\[ w_0 = 4e^{i\frac{2\pi}{3}} = 4 \left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right) = 4 \left( -\frac{1}{2} + i\frac{\sqrt{3}}{2} \right) = -2 + 2i\sqrt{3} \]
\item \textbf{Case $k=1$ (Real Root):}
\[ w_1 = 4e^{i\frac{6\pi}{3}} = 4e^{i2\pi} = 4(\cos 2\pi + i \sin 2\pi) = 4(1 + 0i) = 4 \]
\item \textbf{Case $k=2$ (Conjugate Root):}
\[ w_2 = 4e^{i\frac{10\pi}{3}} = 4 \left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right) = 4 \left( -\frac{1}{2} - i\frac{\sqrt{3}}{2} \right) = -2 - 2i\sqrt{3} \]
\end{itemize}
\subsection*{Conclusion}
The "two answers" usually cited are the **Real Root ($4$)** used in elementary algebra and the **Principal Value ($-2 + 2i\sqrt{3}$)** used by computational engines like WolframAlpha.
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