Calculate ratio $\frac{p^{3n}}{(\frac{p}{2})^{3n}}$

Question

How do I calculate this ratio? I do not know even where to begin.

$$\frac{p^{3n}}{(\frac{p}{2})^{3n}}$$

Thanks

Answer 1

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 $$ \frac{p^{3n}}{\frac{p^{3n}}{2}} = p^{3n}\cdot\frac{2}{p^{3n}} = 2. $$

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

Answer 2

$$\frac{A^k}{\left(\frac{A}2\right)^k}=\frac{A^k}{\frac{A^k}{2^k}}=A^k\cdot\left(\frac{A^k}{2^k}\right)^{-1}=A^k\cdot\frac{2^k}{A^k}=2^k$$