Iknow the latex makes it look really hard, but ! anyone who does gets 15 points.
without using a calculator, order the following numbers from least to greatest:
\begin{align*}
a & = \frac{2^{1/2}}{4^{1/6}}\\
b & = \sqrt[12]{128}\vphantom{dfrac{2}{2} }\\
c& = \left( \frac{1}{8^{1/5}} \right)^2\\
d& = \sqrt{\frac{4^{-1}}{2^{-1} \cdot 8^{-1}}}\\
e& = \sqrt[3]{2^{1/2} \cdot 4^{-1/4}}.\vphantom{dfrac{2}{2}}\en d{align*}give your answer as a list of capital letters separated by commas. for example, if you think that $e < b < c< d< a$, then you would answer e, b,c, d,a.