Question

Compare the following:

`root(4)(12) and root(3)(15)`

Answer 1

`root(4)(12) = 12^(1/4)  "has power" (1)/(4)`

`root(3)(15) = 15^(1/3)  "has power" (1)/(3)`

Now, L.C.M. of 4 and 3 = 12

`root(4)(12) = 12^(1/4) = 12^(3/12) = (12^3)^(1/12) =  (1728)^(1/12)`

`root(3)(15) = 15^(1/3) = 15^(4/12) = (15^4)^(1/12) =  (50625)^(1/12)`

Since 1728 < 50625, we have `(1728)^(1/12) < (50625)^(1/12)`.
Hence, `root(4)(12) < root(3)(15)`.