If `m + (1)/(m) = sqrt3`, then find the simpliest value of (i) `m^(2) + (1)/(m^(2))` and (ii) `m^(3) + (1)/(m^(3))`:

Explanation

(i) `m^(2) + (1)/(m^(2)) = (m)^(2) + ((1)/(m))^(2)`
`= (m + (1)/(m))^(2) -2.m. (1)/(m) = (sqrt3)^(2) -2 = 3 -2 =1`
`:. m^(2) + (1)/(m^(2)) = 1`
(ii) `m^(3) + (1)/(m^(3)) = (m)^(3) + ((1)/(m))^(3) = (m + (1)/(m))^(3) -3.m. (1)/(m) (m + (1)/(m))`
`= (sqrt3)^(3) -3 xx sqrt3 = 3 sqrt3 - 3 sqrt3 = 0`
`:. m^(3) + (1)/(m^(3)) = 0`