Transforming logical circuit in NANDs and ORs
You really should show some work in your question. You don't, except that you do know that you optimized the gate diagram's resulting boolean expression. I suppose that's good enough for now. In any case, others may benefit. So I'll use your question as a foil of sorts.

If you have NANDs, then you want equations of the form: \$\overline{A \cdot B}\$. If you have NORs, then you want equations of the form: \$\overline{A + B}\$. It's not complicated.

You've already worked out the simplified form: \$\overline{B}+A\: C\$. But you need a large bar over all that which covers everything. So to start out you just double-bar everything.

$$\begin{align*}
F &amp;= \overline{B}+A\: C\\\\
&amp;=\overline{\overline{\overline{B}+A\: C}}\\\\
&amp;=\overline{B\cdot \overline{A\: C}}
\end{align*}$$

That is already in perfect form for NANDs, as you should be able to see:







simulate this circuit &ndash; Schematic created using CircuitLab

That's just how easy it is.

Now, can you do the same for NOR? You could follow the above recipe. Or you could just cheat and replace the NANDs with equivalent NORs:







simulate this circuit

So that you get:







simulate this circuit

That one can be optimized a little by replacing two consecutive inverters with a wire:







simulate this circuit