<s>[INST] Simplifying a radical with complex fractions So I understand to simplify this:
$$
\frac{\frac{-3}{2t^4}}{|\frac{1}{2t^3}|\sqrt{\frac{1}{4t^6} - 1}}
$$
I can just multiply
$$
\frac{\frac{-3}{2t^4}}{|\frac{1}{2t^3}|\sqrt{\frac{1}{4t^6} - 1}} \cdot\frac{2t^4}{2t^4}
$$
and get 
$$
\frac{-3}{t\sqrt{\frac{1}{4t^6} - 1}}
$$
But how do you simplify further getting rid of the complex fraction inside the radical?
 [/INST] One may write
$$
\begin{align}
\frac{\frac{-3}{2t^4}}{\left|\frac{1}{2t^3}\right|\sqrt{\frac{1}{4t^6} - 1}}&=\frac{\frac{-3}{2t^4}}{\left|\frac{1}{2t^3}\right|\sqrt{\frac{1-4t^6}{4t^6}}}
\\&=\frac{\frac{-3}{2t^4}}{\left|\frac{1}{2t^3}\right|\frac{\sqrt{1-4t^6}}{\sqrt{4t^6}}}
\\&=\frac{\frac{-3}{2t^4}}{\frac{1}{2\left|t\right|^3}\frac{\sqrt{1-4t^6}}{2|t|^3}}
\\&=\frac{\frac{-3}{2t^4}}{\frac{\sqrt{1-4t^6}}{4t^6}}
\\&=\frac{\frac{-3}{2t^4}\times 4t^6}{\sqrt{1-4t^6}}
\\&=-\frac{6t^2}{\sqrt{1-4t^6}}.
\end{align}
$$
</s>