Is (a*b) or (a*b)* star-free?

Question

Here is the proof of a∗ being star-free:

$\Sigma* = \bar{\emptyset} $

$ A∗= \overline{Σ∗(Σ∖A)Σ∗} $

Would this be a proof for $a * b$? : $ A∗B= \overline{Σ∗(Σ∖A)Σ∗(Σ∖B)} $

For $(A * B )*$ it seems more complicated if not impossible, but is that proofable?

Answer 1

$a^*b = (a^*)\cdot b$

$(a^*b)^* = \{\varepsilon\} \cup \{a,b\}^* b$