This is what I did:
a. let $|A|=α, |B|=β, |C|=γ. |B|≤|C|$ and therefore there is an injection $f: B \to C$, such that $f(b)=c$ for some $b \in B, c \in C.$ Upgrading this function to $f'(a,b)=((id)a,f(b))=(a,c)$, we get an injection from $A\times B$ to $ A\times C$ proving $α · β ≤ α · γ$.
b. Knowing any function defines a unique set of ordered pairs of the form $(x,(f(x))$, the set of functions from $A$ to $B$ and $C$ to $B$ is equivalent to the set of possible sets of ordered pair of the form $(b,a)$ and $(c,a).$ Let $g:B \to A.$ Then $g$ can be written that way: $(b,g(b)).$ Now let us take a function $h: A^B \to A^C$ such that for every $g: B \to A, h(b,g(b))=(f(b),g(b)).$ This is of course an injection, proving $α^ β ≤ α^ γ. $
I would really like your corrections. Thanks.