# Evaluate :- $\frac{(2020^2 - 20100)(20100^2 - 100^2)(2000^2 + 20100)}{10(2010^6 - 10^6)}$

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?

• A first step is to recognize that there are many factors of $10$ here, which allows you to immediately cancel $10^7$. Also in the first factor, the fact that $202-201=1$ helps. Commented Aug 8, 2020 at 7:50
• Wolfram gives $10$ FYI Commented Aug 8, 2020 at 7:54
• Express the numbers as powers of ten. Then try solving. You may find it easier. Commented Aug 8, 2020 at 8:03
• Some motivation on how you could have come up with the solution yourself. Observe that $2010+10=2020$ and also $2010 \cdot 10 = 20100$, with the help of which you could transform the whole expression in terms of only these two numbers. Commented Aug 8, 2020 at 8:38
• True , that's exactly what @Michael Rozenburg has done in his answer, and I should have thought of that myself . Commented Aug 8, 2020 at 8:39

Let $$2010=x$$ and $$10=y$$.

Thus, for our expression we obtain: $$\frac{((x+y)^2-xy)(x^2y^2-y^4)((x-y)^2+xy)}{y(x^6-y^6)}=$$ $$=\frac{(x^2+xy+y^2)y^2(x^2-y^2)(x^2-xy+y^2)}{y(x^6-y^6)}=$$ $$=\frac{y(x^2-y^2)(x^4+x^2y^2+y^4)}{x^6-y^6}=y=10.$$

• Oh I got it , this easy substitution helps a lot . Thank you ! Commented Aug 8, 2020 at 8:33
• @Souradip Das You are welcome! Commented Aug 8, 2020 at 8:33