arrange the following in descending order of magnitude:8th root of 90,4th root of10,sq. root of6 . at last plz explain it also in words.:)

Question

arrange the following in descending order of magnitude:8th root
of 90,4th root of10,sq root of6 at last plz explain it also in
words :)

Aakash Sharma · Accepted Answer

$\sqrt[8]{90} = (90)^{\frac{1}{8}} \sqrt[4]{10} = (10)^{\frac{1}{4}} = ((10^{2})^{\frac{1}{2}})^{\frac{1}{4}} = (100)^{\frac{1}{8}} \sqrt[2]{6} = (6)^{\frac{1}{2}} = ((6^{4})^{\frac{1}{4}})^{\frac{1}{2}} = (1296)^{\frac{1}{8}}$

Now,

1296 > 100 > 90

$\Rightarrow (1296)^{\frac{1}{8}} > (100)^{\frac{1}{8}} > {(90)}^{\frac{1}{8}}$

Hence required descending order is

$\sqrt[2]{6}, \sqrt[4]{10}, \sqrt[8]{90}$