Why the hexadecimal numbers can be converted directly into binary numbers so cleanly? Suppose we have F9 hex. If we want to convert it into binary, we just replace the hexadecimal numbers with their corresponding binary numbers. Like 9 has1001 in binary and F has 1111 in binary. By combining it becomes 1111 1001. But why does it give the right answer when we convert both into decimal number. Why we can't convert any base system into another base system which does not come in its power? As we form a base-3 number system  and a base-9 number system. As 9=3^2 so two digits of base-3 can be represented by a single base-9 digit. But why this happens?

Hint: a sequence of digits $a_n \dots a_0$ in base $g$ corresponds to the natural number $ a_n \cdot g^n + \dots + a_0 \cdot g^0.$ What happen if we consider this number in a base $h$ such that $g=h^k$?
Anyway, even with $2$ and $16$ one needs to be careful: for instance, $F=1111$ and $7=111$ in base $2$, but $F7 = 11110111$ and not $1111111$. Can you see why this happens?
