Comparing 5 and 8

We have $(2.2)^{2}=4.84<5$ , so that $\sqrt{5}>2.2$ . Hence $\sqrt [ 4 ]{ 5 } >\sqrt{2.2}>1.4$ , as $(1.4)^{2}=1.96<2.2$ . Therefore $\sqrt [ 3 ]{ 5 } >\sqrt [ 4 ]{ 5 } >1.4$ . Adding we get

$\sqrt { 5 } +\sqrt [ 3 ]{ 5 } +\sqrt [ 4 ]{ 5 } >2.2+1.4+1.4=5$

We observe that $2^{3}<2^{2}$ (this gives us $\sqrt{8}<2$ ), $\sqrt[3]{8}=2$ and $\sqrt[4]{8}<\sqrt[3]{8}=2$ . Thus

$\sqrt { 8 } +\sqrt [ 3 ]{ 8 } +\sqrt [ 4 ]{ 8 } <2+2+2=6<8$