Insert three irrational numbers between $2\sqrt{3}$ and $2\sqrt{5}$ , and arrange in descending order.

Give an example each of two different irrational numbers , whose

(i) sum is an irrational number.

(ii) product is an irrational number.

Give an example of two different irrational numbers , a and b where $\dfrac{a}{b}$ is a rational number.

In each case, state whether the following numbers are rational or irrational. If they are rational and expressed in the form $\dfrac{p}{q}$, where p and q are coprime integers, then what can you say about the prime factors of q?

$\begin{matrix} \text{(i)} & 279.034 \\[1.5em] \text{(ii)} & 76.\overline{17893} \\[1.5em] \text{(iii)} & 3.010010001… \\[1.5em] \text{(iv)} & 39.546782 \\[1.5em] \text{(v)} & 2.3476817681… \\[1.5em] \text{(vi)} & 59.120120012000… \\[1.5em] \end{matrix}$