How do I calculate this ratio? I do not know even where to begin.
$$\frac{p^{3n}}{(\frac{p}{2})^{3n}}$$
Thanks
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The edits here have changed the question. – Mark Bennet Mar 12 '14 at 09:44
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1@MarkBennet OP changed it himself. view the edit history – Guy Mar 12 '14 at 09:45
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1I made a mistake in the question, sorry. I said that in the comment description. – S0rin Mar 12 '14 at 09:46
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1@Sabyasachi Thanks, I was trying to work out what had happened, and making a mess of it. – Mark Bennet Mar 12 '14 at 09:47
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1Note that $\cfrac {A^k}{B^k}=\left(\cfrac AB\right)^k$ – Mark Bennet Mar 12 '14 at 09:49
2 Answers
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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 $$ \frac{p^{3n}}{\left(\frac{p}{2}\right)^{3n}} = \frac{p^{3n}}{\left(\frac{p^{3n}}{2^{3n}}\right)} = p^{3n} \cdot \frac{2^{3n}}{p^{3n}} = 2^{3n}. $$
Christoph
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2@Sabyasachi For this kind of question I like to stress every single step, since this question is about the very nature of dealing with fractions. – Christoph Mar 12 '14 at 09:50
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@Christoph thank you for the help. If I want to read more on this, what should I search for? Algebraic ratios? – S0rin Apr 16 '14 at 13:52
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$$\frac{A^k}{\left(\frac{A}2\right)^k}=\frac{A^k}{\frac{A^k}{2^k}}=A^k\cdot\left(\frac{A^k}{2^k}\right)^{-1}=A^k\cdot\frac{2^k}{A^k}=2^k$$
Did
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2@vonbrand Nothing, A is larger, hence easier to read. But changing p to A and 3n to k is also a way to suggest that the result is general (the next step would to replace 2 by B). – Did Mar 12 '14 at 09:49
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@vonbrand Asking for the author's intent (and receiving an answer) is one thing. Deciding to modify the answer to follow your personal whim is another. Please do not do that. – Did Mar 12 '14 at 09:55