We consider three classes of $N$-functions: $(\Delta')$, the class of functions satisfuing the $\Delta'$ condition, $(\Delta_2)$, the class of functions satisfuing the $\Delta_2$ condition, and $(M_\Delta)$, the class of functions $M(u)$ satisfying the condition: $\lim\limits_{u\to\infty}\ln M(u)/\ln u =p<\infty$. We establish the connection between the class of powers and the class of $N$-functions $M(u)$ which belong to the class $(\Delta')$ together with their complementary functions and we also establish the connections between the classes $(\Delta')$, $(M_\Delta)$ and $(\Delta_2)$.