X1. Prove that if x is algebraic and r is rational, then x+r and rx are both algebraic.
X2. Prove that if x is algebraic and positive, then for every positive integer n, x^(1/n) is algebraic.
X3. Prove that if r and s are positive rational numbers, then r^(1/2)+s^(1/2) is algebraic.
Hints for #3 in the book:
There are three hints here. After reading one hint, try and see if you can do the problem before reading the next hint.
(i) Use limit comparison test.
(ii) Compare with the harmonic series.
(iii) Use L'Hospital's rule.