(1+x)^2/x^6 doesn't simplify?

[Pengwuino]

Ok I got an equation here...
$$2\sqrt {\frac{{(1 + 2t^4 )^2 }}{{t^6 }}}$$
is the equation (sorry for misleading title but its the same concept).
Now I was always under the assumption that the top square could be removed as long as you reduce the bottom to $$t^3$$ since $$t^3$$ squared is $$t^6$$ and that you can cancel out squares like that. I guess I'm wrong? I'm running some examples in my mind and I'm kinda realizing you can't do it... but i feel like I've always thought it was true for soem reason.
I also did a simplification through mathematica and found out that the square root actually allows you to remove the $$^3$$ to get to the $$t^3$$.
The first question I'm asking is: without that square root... would I be able to simplify?
The 2nd question is exactly why I am able to use that square root to simplify. Sorry if its confusing...

 

[TD]

Well you can get rid off the square root, since

$$\sqrt {\frac{{a^2 }}<br /> {{b^2 }}} = \sqrt {\left( {\frac{a}<br /> {b}} \right)^2 } = \left| {\frac{a}<br /> {b}} \right|$$

 

[Pengwuino]

Yah I was under the impression that even without that square root, it could turn into a/b

 

[Knavish]

Just remember that the quantity has to remain positive. So the simplification of the original equation is:
$$|{1/t^3 + 2t}|$$

 

[FrogPad]

If you want to see the reduction broken down then this is what you should do.
$$2\sqrt{\frac{(1+t^4)^2}{t^6}} =2\left(\frac{(1+t^4)^{2\frac{1}{2}}}{t^6}\right)^{\frac{1}{2}}$$
$$=2\frac{(1+t^4)^{2\frac{1}{2}}}{t^{6\frac{1}{2}}}=2\frac{1+t^4}{t^3}$$
$$=2t^{-3}+2t^{-3+4}=|2(t^{-3}+t)|$$

which almost yields the same result as knavish. I think he is missing a factor of 2 somewhere in there as

p.s. sorry I didn't break it down with the absolute value signs, but those were already explained pretty well... and for some reason I like to add them afterwards. bad habit i guess