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.

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

$\begin{matrix} \text{(i)} & \dfrac{13}{3125} \\[1.5em] \text{(ii)} & \dfrac{17}{8} \\[1.5em] \text{(iii)} & \dfrac{23}{75} \\[1.5em] \text{(iv)} & \dfrac{6}{15} \\[1.5em] \text{(v)} & \dfrac{1258}{625} \\[1.5em] \text{(vi)} & \dfrac{77}{210} \\[1.5em] \end{matrix}$

Write the decimal expansion of each of the following numbers and say what kind of decimal expansion each has:

$\begin{matrix} \text{(i)} & \dfrac{36}{100} \\[1.5em] \text{(ii)} & 4\dfrac{1}{8} \\[1.5em] \text{(iii)} & \dfrac{2}{9} \\[1.5em] \text{(iv)} & \dfrac{2}{11} \\[1.5em] \text{(v)} & \dfrac{3}{13} \\[1.5em] \text{(vi)} & \dfrac{329}{400} \\[1.5em] \end{matrix}$

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

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