1.$\displaystyle \sqrt[4]{8x^5}$
$\displaystyle 2|x|\sqrt{x}$
2.$\displaystyle \sqrt{49x^6}$
$\displaystyle 7|x^3| or |7x^3$|? which one is valid?
3.$\displaystyle \sqrt{x^3}$
$\displaystyle |x|\sqrt{x}$?
Both. But obviously the first is simpler.
And you should note that:
1. the domain of $\displaystyle \sqrt{x^n}$ is $\displaystyle x \geq 0$ if n is odd (why?) and so |x | = x under such circumstances.
2. the domain of $\displaystyle \sqrt{x^n}$ is $\displaystyle R$ if n is even (why?).