Express the following numbers in the form $\dfrac{p}{q}$, where p and q are both integers and q ≠ 0.

$\begin{matrix} \text{(i)} & 0.\overline{3} \\[1.5em] \text{(ii)} & 5.\overline{2} \\[1.5em] \text{(iii)} & 0.404040… \\[1.5em] \text{(iv)} & 0.4\overline{7} \\[1.5em] \text{(v)} & 0.1\overline{34} \\[1.5em] \text{(vi)} & 0.\overline{001} \\[1.5em] \end{matrix}$

Write the denominator of the rational number $\dfrac{257}{5000}$ in the form 2^m × 5ⁿ where m, n are non-negative integers. Hence, write its decimal expansion without actual division.

Write the decimal expansions of the following numbers which have terminating decimal expansions:

$\begin{matrix} \text{(i)} & \dfrac{17}{8} \\[1.5em] \text{(ii)} & \dfrac{13}{3125} \\[1.5em] \text{(iii)} & \dfrac{7}{80} \\[1.5em] \text{(iv)} & \dfrac{6}{15} \\[1.5em] \text{(v)} & \dfrac{2^2 \times 7}{5^4} \\[1.5em] \text{(vi)} & \dfrac{237}{1500} \\[1.5em] \end{matrix}$

Classify the following numbers as rational or irrational:

$\begin{matrix} \text{(i)} & \sqrt{23} \\[1.5em] \text{(ii)} & \sqrt{225} \\[1.5em] \text{(iii)} & 0.3796 \\[1.5em] \text{(iv)} & 7.478478… \\[1.5em] \text{(v)} & 1.101001000100001… \\[1.5em] \text{(vi)} & 345.0\overline{456} \\[1.5em] \end{matrix}$