Evaluate :- $\frac{(2020^2 – 20100)(20100^2 – 100^2)(2000^2 + 20100)}{10(2010^6 – 10^6)}$
What I Tried :- I couldn't think of any ways to factorise this expression . The denominator can be written as $10(2016^3 – 10^3)(2016^3 + 10^3)$ , but I can't understand how it will help here . I have absolutely no idea how to factorise the numerator except that it can be $(2020^2 – 20100)(20100 – 100)(20100 + 100)(2000^2 + 20100)$, other than that I got no idea, and it seems to me the only way to get it is to open the brackets , which will contain a lot of calculations .
Wolfram Alpha gives the answer to be $10$ . But I am looking for some clever way so that this expression gets factorised and I can get my answer in less calculations .
Can anyone help?
Best Answer
Let $2010=x$ and $10=y$.
Thus, for our expression we obtain: $$\frac{((x+y)^2-xy)(x^2y^2-y^4)((x-y)^2+xy)}{y(x^6-y^6)}=$$ $$=\frac{(x^2+xy+y^2)y^2(x^2-y^2)(x^2-xy+y^2)}{y(x^6-y^6)}=$$ $$=\frac{y(x^2-y^2)(x^4+x^2y^2+y^4)}{x^6-y^6}=y=10.$$