The greatest of the numbers $1,2^{\frac{1}{2}},3^{\frac{1}{3}},4^{\frac{1}{4}},5^{\frac{1}{5}},6^{\frac{1}{6}}$  and $7^{\frac{1}{7}}$  is

$f\left(x\right)=x^{\frac{1}{x}}$  is increasing on $\left(0,e\right)$ and is decreasing on $\left(e,\infty \right)$  As $e<3<4<5<6<7$ $\Rightarrow$  max. of $3^{\frac{1}{3}},4^{\frac{1}{4}},5^{\frac{1}{5}},6^{\frac{1}{6}},7^{\frac{1}{7}}$  is $3^{\frac{1}{3}}$  Also $1<2<e\Rightarrow$  max. of 1 and $2^{\frac{1}{2}}$  is $2^{\frac{1}{2}}$  But $2^{\frac{1}{2}}=4^{\frac{1}{4}}$

$\therefore$  greatest is $3^{\frac{1}{3}}$