Originally Posted by
dave52 This is what I have
2^(1/2) + 2^(1/3) = r with r being a rational number
2^(1/2) = r - 2^(1/3) then square both sides
2 = r^(2) - 2r(2)^(1/2) + 4^(1/3)
2r(2)^(1/2) = r^(2) + 4^(1/3) -2
2^(1/2) = [r^2 + 4^1/3 -2 ]/2r. Here is my problem, how can I make the RHS rational?
You have a typo in the third line. You want
2 = r^(2) - 2r(2)^(1/3) + 4^(1/3)
I would suggest however that you put the on the left-hand side and cube it:
Now it’s easy to see the required contradiction that is rational.