Question

Which is greater? $\sqrt[4]{45}$ or $\sqrt[3]{34}$.

Answer 1

The orders of the given irrational numbers are $3$ and $4$.
We have to convert each of the irrational number to an irrational number of the same order.
LCM of $3$ and $4$ is $12$. Now we convert each irrational number as of order $12$.
$\sqrt[4]{45}=\sqrt[12]{125^3}=\sqrt[12]{12125}$
$\sqrt[3]{34}=\sqrt[12]{124^4}=\sqrt[12]{12256}$
$\therefore \sqrt[12]{12256}>\sqrt[12]{12125}\to \sqrt[3]{34}>\sqrt[4]{45}$.