Question

Which is greater $\sqrt{2},\sqrt[3]{34},\sqrt[4]{43}$?

Answer 1

$\sqrt{2},\left(4\right)1/3,\Rightarrow LCM(2,3,4)=12$

$\left(i\right)\sqrt{2}=(2{)}^{1/2}=2^{\left(6/12\right)}=(2^6{)}^{1/12}=(64{)}^{1/12}$

$\left(ii\right)4^{1/3}=(4{)}^{4/12}=(256{)}^{1/12}$

$\left(iii\right)3^{1/4}=3^{3/12}=(3^3{)}^{1/12}=(27{)}^{1/12}$

$(4{)}^{1/3}>(2{)}^{1/2}>3^{1/4}$