Waldronjw

2022-07-04

Simplifying a radical with complex fractions
So I understand to simplify this:
$\frac{\frac{-3}{2{t}^{4}}}{|\frac{1}{2{t}^{3}}|\sqrt{\frac{1}{4{t}^{6}}-1}}$
I can just multiply
$\frac{\frac{-3}{2{t}^{4}}}{|\frac{1}{2{t}^{3}}|\sqrt{\frac{1}{4{t}^{6}}-1}}\cdot \frac{2{t}^{4}}{2{t}^{4}}$
and get
$\frac{-3}{t\sqrt{\frac{1}{4{t}^{6}}-1}}$
But how do you simplify further getting rid of the complex fraction inside the radical?

haingear8v

Expert

$\frac{\frac{-3}{2{t}^{4}}}{|\frac{1}{2{t}^{3}}|\sqrt{\frac{1}{4{t}^{6}}-1}}\frac{\sqrt{4{t}^{6}}}{\sqrt{4{t}^{6}}}$ first step will be to get rid of the fraction inside the radical. Note that $\sqrt{4{t}^{6}}=|2{t}^{3}|$
$\frac{\frac{-3}{2{t}^{4}}|2{t}^{3}|}{|\frac{1}{2{t}^{3}}|\sqrt{1-4{t}^{6}}}\frac{\sqrt{1-4{t}^{6}}}{\sqrt{1-4{t}^{6}}}$ Then we get the radical outside of the numerator.
$\frac{\frac{-3}{2{t}^{4}}|2{t}^{3}|\sqrt{1-4{t}^{6}}}{|\frac{1}{2{t}^{3}}|\left(1-4{t}^{6}\right)}\frac{|2{t}^{3}|}{|2{t}^{3}|}$ Now I am taking on what you did in the first step. I think it is best to attack the messiest parts first.
$\frac{\frac{-3}{2{t}^{4}}\left(4{t}^{6}\right)\sqrt{1-4{t}^{6}}}{\left(1-4{t}^{6}\right)}\phantom{\rule{0ex}{0ex}}\frac{-6{t}^{2}\sqrt{1-4{t}^{6}}}{\left(1-4{t}^{6}\right)}$

gaiaecologicaq2

Expert

$\begin{array}{rl}\frac{\frac{-3}{2{t}^{4}}}{|\frac{1}{2{t}^{3}}|\sqrt{\frac{1}{4{t}^{6}}-1}}& =\frac{\frac{-3}{2{t}^{4}}}{|\frac{1}{2{t}^{3}}|\sqrt{\frac{1-4{t}^{6}}{4{t}^{6}}}}\\ & =\frac{\frac{-3}{2{t}^{4}}}{|\frac{1}{2{t}^{3}}|\frac{\sqrt{1-4{t}^{6}}}{\sqrt{4{t}^{6}}}}\\ & =\frac{\frac{-3}{2{t}^{4}}}{\frac{1}{2{|t|}^{3}}\frac{\sqrt{1-4{t}^{6}}}{2|t{|}^{3}}}\\ & =\frac{\frac{-3}{2{t}^{4}}}{\frac{\sqrt{1-4{t}^{6}}}{4{t}^{6}}}\\ & =\frac{\frac{-3}{2{t}^{4}}×4{t}^{6}}{\sqrt{1-4{t}^{6}}}\\ & =-\frac{6{t}^{2}}{\sqrt{1-4{t}^{6}}}.\end{array}$

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