Prove that the following languages are not regular.

$($b$)L=\{a^n{b}^l{a}^k$:$kn+l\}.$

$($c$)L=\{\begin{array}{c}{a}^n{b}^l{a}^k\text{:}n=l\end{array}$ or $\{$:$lk\}.$

$($d$)L=\{a^n{b}^l$:$nl\}.$

$($e$)L=\{w$:$n_a(w)n_b(w)\}.$

$($f$)L=\{ww$:win$\{a,b{\}}^{**}\}.$