5. For each part, show that the language is regular, or use the pumping lemma to show that the language is
nonregular. You must justify your answers to receive full credit. 20 points
a) $L_5 = \{w0w^R | w \in \{0, 1\}^*\}$ where $s^R$ is the reverse of string $s$
b) $L_6 = \{0^{n^3} | n \ge 0\}$
c) $L_7 = \{0^{2^n} | n \ge 0\}$
d) $L_8 = \{0^i 1^j 0^k | i < j < k \text{ and } i, j, k > 0\}$