Question

Prove that a=b=c=0 when a+bp^1/3+cp^2/3=0

Answer 1

Answer:

a, b, c and p are rational.  p is not a perfect cube.

 so p¹/³ and p²/³ are not rational numbers. Then p is not 1 or 0.

p²/³ = p¹/³ * p¹/³   and so they are not equal.

        LHS = a + b p¹/³ + c p²/³ = 0       --- (1)  Given p is not a perfect cube. p is not 0 or 1. Also  are irrational. Multiply (1) by p^1/3 to get:  Substitute the value of c in (1) to get:  So p^1/3 is imaginary. It is a contradiction as p is a rational number.  Given quadratic isn't valid. So a = b = c = 0