Mathematics

Problem

Study the expression

E=\sqrt{2}+\sqrt{18}-\sqrt{9}+\frac{3}{4}.

(i) Simplify

E

as far as possible. (ii) Determine whether the simplified value of

E

is rational or irrational. Give a clear justification for your answer. (iii) Let

a

be a positive integer that is not a perfect square and let

b=k^{2}a

for some positive integer

k

. Prove that

\sqrt{a}+\sqrt{b}

is always an irrational number.