Question

$\left(6+\sqrt{27}\right)-\left(3+\sqrt{3}\right)+\left(1+2\sqrt{3}\right)$ when simplified is
(a) positive and irrational
(b) positive and rational
(c) negative and irrational
(d) negative and rational

Answer 1

$$\begin{gathered} \left(6+\sqrt{27}\right)-\left(3+\sqrt{3}\right)+\left(1+2\sqrt{3}\right) \\ =6+\sqrt{3\times 3\times 3}-\left(3+\sqrt{3}\right)+\left(1+2\sqrt{3}\right) \\ =6+3\sqrt{3}-3-\sqrt{3}+1+2\sqrt{3} \\ =4+4\sqrt{3} \end{gathered}$$
Thus, the given expression when simplified is positive and irrational.

Hence, the correct answer is option (a).