Question 414466
<font face=”Garamond” size=”+2”>


You have:


*[tex \LARGE\ \ \ \ \ \ \ \ \ \ \frac{\sqrt[3]{q\ -\ 1}}{\sqrt[4]{q\ -\ 1}}] 


The easiest way to do this so it is clearly understandable is to write the roots as rational exponents, thus:


*[tex \LARGE\ \ \ \ \ \ \ \ \ \ \frac{\left(q\ -\ 1\right)^{\frac{1}{3}}}{\left(q\ -\ 1\right)^{\frac{1}{4}}}]


Then recall the law of exponents that says:  *[tex \Large \frac{a^m}{a^n}\ =\ a^{m\,-\,n}]


Calculate *[tex \Large \frac{1}{3}\ -\ \frac{1}{4}\ =\ \frac{1}{12}].


So:


*[tex \LARGE\ \ \ \ \ \ \ \ \ \ \frac{\sqrt[3]{q\ -\ 1}}{\sqrt[4]{q\ -\ 1}}\ =\ \left(q\ -\ 1\right)^{\frac{1}{12}}\ =\ \sqrt[12]{q\ -\ 1}]




John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
<div style="text-align:center"><a href="http://outcampaign.org/" target="_blank"><img src="http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png" border="0" alt="The Out Campaign: Scarlet Letter of Atheism" width="143" height="122" /></a></div>
</font>