What does $\overline{L}^*$ signify?
For any language&nbsp;$L$ over alphabet&nbsp;$\Sigma$,


$\overline{L} = \Sigma^*\setminus L$ is the complement of&nbsp;$L$: the set of $\Sigma$-strings not in&nbsp;$L$;
$L^*$ is the Kleene closure of&nbsp;$L$: the set of strings&nbsp;$w\in\Sigma^*$ such that we can write $w=w_1\dots w_k$ for some $k\geq 0$ such that each $w_i\in L$.  For example, if $L$&nbsp;is the set of English words, then correcthorsebatterystaple is in&nbsp;$L^*$.


Now, $\overline{L}^*$ is something-star, where "something" is $\overline{L}$. That is, it's the Kleene closure of the complement of&nbsp;$L$: the set of all strings $w\in\Sigma^*$ that can be written $w_1\dots w_k$ such that each $w_i$&nbsp;is not in&nbsp;$L$. Essentially, we read left-to-right, so you parse "$\overline{L}$" before you reach the star.

Conversely, $\overline{L^*}$ is something-complement, where "something" is&nbsp;$L^*$, the Kleene closure of&nbsp;$L$. That is, it is the set of all strings that cannot be written $w_1\dots w_k$ where each $w_i\in L$.

These are not necessarily the same languages. For example, consider $\Sigma=\{1\}$ and $L=\{11\}$.  Then $1\in\overline{L}$, so $\overline{L}^*=\Sigma^*$. However, $$\overline{L^*} = \overline{\{1^i\mid i\text{ is even}\}} = \{1^i\mid i\text{ is odd}\}\neq\Sigma^*\,.$$