Question

x+1/root x =3 then what will be the value of x cube + 1/x cube

Answer 1

x=2+∫3   [given]

x+1/x =2+∫3+1/2+∫3

         =[2+∫3][2+∫3]+1/2+∫3   [taking LCM ]

         =[2+∫3]2+1/2+∫3

         =4+3+[4×∫3]+1/2+∫3

         =8+4∫3/2+∫3

         =[8+4∫3/2+∫3]×[2-∫3/2-∫3]    [by rationalising the denominator]

         =[8+4∫3][2-∫3]/[2]2-[∫3]2

         =16-8∫3+8∫3-4[∫3]2

            =16-12

x+1/x=4

[x+1/x]3=43     [cubing both side]

x3+1/x3+3×x×1/x[x+1/x]=64     [using {x+y}3]

x3+1/x3+3×4=64     [substituting the value of x+1/x]\

x3+1/x3=64-12

x3+1/x3=52 [Ans]