One root of equation ax^2 + bx + c = 0 is equal to nth power of the other root. Show that (a^n .c)^(1/n+1) + (a.c^n)^(1/n+1) + b = 0

The equation ax^2 + bx + c = 0 has roots such that one roots is equal to nth power of the other. We have to prove (a.c^n)^(1/ n+1) + (a^n .c)^(1/ n+1) + b = 0

We can assume the roots to be r and r^n. By applying sum and product of roots formulae on equation ax^2 + bx + c= 0, we'll get

Product of roots = c/a

=> r . r^n = c/a

=> r^(n+1)= c/a

=> r = (c/a)^(1/ n+1)

Sum of roots = -b/a

=> r + r^n = -b/a

Now, by substituting the value of r we found above,

=> (a^n .c)^(1/n+1) + (a.c^n)^(1/n+1) + b = 0 

Hence Proved.