Calculate ratio $\frac{p^{3n}}{(\frac{p}{2})^{3n}}$ How do I calculate this ratio? I do not know even where to begin. 
$$\frac{p^{3n}}{(\frac{p}{2})^{3n}}$$
Thanks

Regarding the original question:
$$
\frac{p^{3n}}{\frac{p^{3n}}{2}} = 
\frac{p^{3n}}{\frac{p^{3n}}{2}} \cdot 1 =
\frac{p^{3n}}{\frac{p^{3n}}{2}} \cdot \frac{2}{2} =
\frac{p^{3n}\cdot 2}{\frac{p^{3n}}{2} \cdot 2} =
\frac{p^{3n}\cdot 2}{p^{3n}\cdot 1} =
\frac{p^{3n}}{p^{3n}} \cdot \frac 2 1 =
1\cdot \frac 2 1 = 2.
$$
In general we have
$$
\frac{A}{\frac p q} = \frac{A\cdot q}{\frac p q \cdot q} = A\cdot\frac{q}{p},
$$
so dividing by $\frac p q$ is the same as multiplying by $\frac q p$. Using this you also get
$$
\frac{p^{3n}}{\frac{p^{3n}}{2}} = p^{3n}\cdot\frac{2}{p^{3n}} = 2.
$$

For your updated question we use $\left(\frac{p}{q}\right)^k = \frac{p^k}{q^k}$ to obtain
$$
\frac{p^{3n}}{\left(\frac{p}{2}\right)^{3n}} = 
\frac{p^{3n}}{\left(\frac{p^{3n}}{2^{3n}}\right)} = 
p^{3n} \cdot \frac{2^{3n}}{p^{3n}} = 2^{3n}.
$$
