Question

how did the ideas of the enlightenment and increasing commercial development lead to the rapid increase in the size of the british empire

Answer 1

Here is the answer. How the idea of the enlightenment and  increasing commercial development led to the rapid increase in the size of the British empire is that it made people feel more justified in taking large areas of land for their country, since this was the "rational" and pragmatic move in the age of science and exploration.  Hope this helps.