For each question, select the proper option from four options given, to make the statement true : (Final answer only)

$0.3 \overline{6}=$

Simplify the following:

$\frac{2 \sqrt{3}}{3}-\frac{\sqrt{3}}{6}$

Let $x$ be rational and $y$ be irrational. Is xy necessarily irrational? Justify your answer by an example.

If $\sqrt{2}=1.414, \sqrt{3}=1.732,$ then find the value of $\frac{4}{3 \cdot \sqrt{3}-2 \cdot \sqrt{2}}+\frac{3}{3 \cdot \sqrt{3}+2 \cdot \sqrt{2}}$

prove that

$\frac{x^{a(b-c)}}{x^{b(a-c)}} \div\left(\frac{x^{b}}{x^{a}}\right)^{c}=1$

Find four rational numbers between $\frac{2}{9}$ and $\frac{2}{7}$