Non-context free languages with word degree [duplicate]

I have stumbled across these 2 problems

1. $L_1= \{\alpha \mid w \in \{a,b\}^* | \alpha$ has exactly 2 b's$\}$ ,prove that $L =\{ \alpha^n | \alpha ∈ L_1 ,n \ge 0 \}$ is not context free

2. Given : $P= \{\alpha \mid w \in \{a,b,c\}^* | \alpha$ is palindrom with even length $\}$ and $L_2= \{\beta b^n| b$ ∈ $P^n$ and $n ∈ \mathbb{N}$} prove that $L_2$ is context free.

In both problems we have degree $n$ of a word with length >1,usually in such problems how do we cope with that ?Also any suggestions,tricks or hints how to proceed in these 2 problems or in general with degree $n$ will be very helpful