Question

Write the following in descending order:

`sqrt(2), root(3)(5) and root(4)(10)`

Answer 1

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

`root(3)(5) = 5^(1/3)  "has power" (1)/(3)`

`root(4)(10) = 10^(1/4)  "has power" (1)/(4)`

Now, L.C.M. of 2, 3 and 4 = 12

∴ `sqrt(2) = 2^(1/2) = 2^(6/12) = (2^6)^(1/12) = (64)^(1/12)`

`root(3)(5) = 5^(1/3) = 5^(4/12) = (5^4)^(1/12) = (625)^(1/12)`

`root(4)(10) = 10^(1/4) = 10^(3/12) = (10^3)^(1/12) = (1000)^(1/12)`

Since, 1000 > 625 > 64, we have `(1000)^(1/12) > (625)^(1/12) > (64)^(1/12)`.
Hence, `root(4)(10) > root(3)(5) > sqrt(2)`.