Greatest Number.

Here I will give the solution. first of all we to equalise the nth roots. ie. By taking its LCM(4,5,10). Which is 20. $3^\frac{1}{4} , 4^\frac{1}{5} , 12^\frac{1}{10}$ becomes, $( 3^5)^\frac{1}{20} , ie. 243^\frac{1}{20},$ and others become $256^\frac{1}{20} , 144^\frac{1}{20}$ and 1 can be written as $1^\frac{1}{20}$

Now it is clearer and easy to find the answer. It is $4^\frac{1}{5}$ . Hope you understood.

Raise each of them to the power 20 ( Multiple of their powers - 4,5,10,1). And then compare 243,256,144 and 1 respectively for each term.

n th root of any integer is greater than 1