Question

Compare $\sqrt{\sqrt[3]{312}}$ and $\sqrt[3]{3\sqrt[4]{4138}}$.

Answer 1

We write
$\sqrt{\sqrt[3]{312}}={\left(\right(12{)}^{1/3})}^{1/2}=(12{)}^{1/6}$,
Similarly, we get
$\sqrt[3]{3\sqrt[4]{4138}}={\left(\right(138{)}^{1/4})}^{1/3}=(138{)}^{1/12}$.
Now the indices are $1/6$ and $1/12$. We can bring the first to the same index:
$(12{)}^{1/6}=({12}^2{)}^{1/12}=(144{)}^{1/12}$.
Now we have two numbers,

$(144{)}^{1/12}$ and $(138{)}^{1/12}$ having the same indices.

Comparing their radicals, we obtain $(144{)}^{1/12}$$>$$(138{)}^{1/12}$
i.e. $\sqrt[3]{3\sqrt[4]{4138}}<\sqrt{\sqrt[3]{312}}$.