The greatest among the numbers $\sqrt[4]{3}, \sqrt[5]{4}, \sqrt[10]{12}, 1$ is

LCM of indices of surds

= LCM of 4, 5 and 10 = 20

$\therefore \sqrt[4]{3}=(3)^{\frac{1}{4}}=3^{\frac{5}{20}}=\left(3^{5}\right)^{\frac{1}{20}}=(243)^{\frac{1}{20}}$

$\sqrt[5]{4}=(4)^{\frac{1}{5}}=\left(4^{4}\right)^{\frac{1}{20}}=(256)^{\frac{1}{20}}$

$\sqrt[10]{12}=(12)^{\frac{1}{10}}$

$=\left(12^{2}\right)^{\frac{1}{20}}=(144)^{\frac{1}{20}}$

$\therefore$ Largest number $\sqrt[5]{4}=(256)^{\frac{1}{20}}$