Answer
$9+3t^5+\dfrac{1}{4}t^{10}$
Work Step by Step
Using $(a+b)^2=a^2+2ab+b^2$ or the square of a binomial, the given expression, $
\left( 3+\dfrac{1}{2}t^5 \right)\left( 3+\dfrac{1}{2}t^5 \right)
$, is equivalent to
\begin{array}{l}\require{cancel}
(3)^2+2(3)\left( \dfrac{1}{2}t^5 \right)+\left( \dfrac{1}{2}t^5 \right)^2
\\\\=
9+3t^5+\dfrac{1}{4}t^{10}
.\end{array}