Answer
$\dfrac{3}{4}t^{3}u^{2}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
\sqrt[4]{\dfrac{81}{256}t^{12}u^{8}}
,$ find a factor of the radicand that is a perfect power of the index. Then extract the root of that factor. Note that all variables are assumed to represent positive real numbers.
$\bf{\text{Solution Details:}}$
Expressing the radicand of the expression above with a factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt[4]{\left( \dfrac{3}{4}t^{3}u^{2}\right)^4}
\\\\=
\dfrac{3}{4}t^{3}u^{2}
.\end{array}