Simplifying a radical with complex fractions

Question

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?

Answer 1

$\frac{\frac{-3}{2t^4}}{|\frac{1}{2t^3}|\sqrt{\frac{1}{4t^6} - 1}} \frac{\sqrt {4t^6}}{\sqrt{4t^6}}$ first step will be to get rid of the fraction inside the radical. Note that ${\sqrt{4t^6}} = |2t^3|$

$\frac{\frac{-3}{2t^4}|2t^3|}{|\frac{1}{2t^3}|\sqrt{1 - 4t^6}} \frac{\sqrt {1-4t^6}}{\sqrt{1-4t^6}}$ Then we get the radical outside of the numerator.

$ \frac{\frac{-3}{2t^4}|2t^3|\sqrt {1-4t^6}}{|\frac{1}{2t^3}|(1 - 4t^6)} \frac{|2t^3|}{|2t^3|}$ Now I am taking on what you did in the first step. I think it is best to attack the messiest parts first.

$ \frac{\frac{-3}{2t^4}(4t^6)\sqrt {1-4t^6}}{(1 - 4t^6)}\\ \frac{-6t^2\sqrt {1-4t^6}}{(1 - 4t^6)}$

Answer 2

One may write $$ \begin{align} \frac{\frac{-3}{2t^4}}{\left|\frac{1}{2t^3}\right|\sqrt{\frac{1}{4t^6} - 1}}&=\frac{\frac{-3}{2t^4}}{\left|\frac{1}{2t^3}\right|\sqrt{\frac{1-4t^6}{4t^6}}} \\&=\frac{\frac{-3}{2t^4}}{\left|\frac{1}{2t^3}\right|\frac{\sqrt{1-4t^6}}{\sqrt{4t^6}}} \\&=\frac{\frac{-3}{2t^4}}{\frac{1}{2\left|t\right|^3}\frac{\sqrt{1-4t^6}}{2|t|^3}} \\&=\frac{\frac{-3}{2t^4}}{\frac{\sqrt{1-4t^6}}{4t^6}} \\&=\frac{\frac{-3}{2t^4}\times 4t^6}{\sqrt{1-4t^6}} \\&=-\frac{6t^2}{\sqrt{1-4t^6}}. \end{align} $$