<s>[INST] Determine the value of $\frac{\sqrt[3]{4\sqrt[3]{4\sqrt[3]{4...}}}}{\sqrt{2\sqrt {2\sqrt{2...}}}}$ Determine the value for 

$$\frac{\sqrt[3]{4\sqrt[3]{4\sqrt[3]{4...}}}}{\sqrt{2\sqrt {2\sqrt{2...}}}}$$

I think the formula $S_\infty =\frac {a}{1-r}$ should be used for this question but I don’t know how to find the first term,a and the common ratio, r.
Please show how to solve this question in the simplest way possible.
 [/INST] Hint:
$$\sqrt{2\sqrt {2\sqrt{2...}}}=2^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots}$$
and
$$\sqrt[3]{4\sqrt[3]{4\sqrt[3]{4...}}}=4^{\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\ldots}$$
and now you can use the geometric series formula.
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