How to evaluate this using algebra? We have $$1.001^6 - 1.001^5$$ How do I evaluate this? Normally I would use algebra to rewrite it, but I don't know how to cleverly rewrite $a^6 - a^5$ to a simpler form.

How about (let $x = 0.001$ for simpler writing) \begin{align*} (1+x)^6 - (1+x)^5 &= (1+x)^5 \cdot (1+x - 1)\\\ &= x \cdot (1+x)^5\\\ &= x^6 + 5x^5 + 10x^4 + 10x^3 + 5x^2 + x\\\ &= 0.001\,005\,010\,010\,005\,001 \end{align*}