Question

Write the following in descending order:

`sqrt(6), root(3)(8) and root(4)(3)`

Answer 1

Since `sqrt(6) = 6^(1/2)  "has power" (1)/(2)`,

`root(3)(8) = 2`

`root(4)(3) = 3^(1/4)  "has power" (1)/(4)`

Now, L.C.M. of 2, 1 and 4 = 4

∴ `sqrt(6) = 6^(1/2) = 6^(2/4) = (6^2)^(1/4) = (36)^(1/4)`

`root(3)(8) = 2 = 2^(4/4) = (2^4)^(1/4) = (16)^(1/4)`

`root(4)(3) = 3^(1/4) = (3^1)^(1/4) = (3)^(1/12)`

Since, 36 > 16 > 3, we have `(36)^(1/4) > (16)^(1/4) > (3)^(1/12)`.
Hence, `sqrt(6) > root(3)(8) > root(4)(3)`.