Greatest of the numbers given To find out the greatest among the number given below:
$3^{1/3}, 2^{1/2}, 6^{1/6}, 1, 7^{1/7}$
I have plotted the following graph using graph plotter which is shown below:

It can be concluded that $3^{1/3}$ is the greatest.
I want to know that is there any other method to find greatest among the such numbers.

Take lcm$(3,2,6,7)=42$
We need to check for $3^{1/3},2^{1/2},6^{1/6},7^{1/7}$
equivalently taking $42$nd power of each  $3^{14},2^{21},6^7,7^6$
Now $2^3<3^2\iff(2^3)^7<(3^2)^7$
Again, $3^{14}-6^7=3^7(3^7-2^7)>0$
and finally $3^7>3^6=729>343=7^3\implies(3^7)^2>(7^3)^2$
