What is the value of $$\frac{2013^3-2\cdot 2013^2\cdot 2014+3\cdot 2013\cdot 2014^2-2014^3+1}{2013\cdot 2014}?$$
What I have tried:
$$\implies\frac{2013^2(2013-2\cdot2014)+2014^2(3\cdot 2013-2014)+1}{2013\cdot 2014}$$ $$\implies\frac{2013^2(-2015)+2014^2(4025)+1}{2013\cdot 2014}$$
I'm not sure what to do next...
Help is appreciated!
Furthermore, if you are nice, could you also help me on this problem(Transferring bases of numbers.) too?
Thanks!
Max0815
You can complete the cube of the difference.
$$ =\frac{2013^3-3\cdot2013^2\cdot2014+3\cdot2013\cdot2014^2-2014^3+2013^2\cdot2014+1}{2013\cdot2014} $$ $$ =\frac{(2013-2014)^3+2013^2\cdot2014+1}{2013\cdot2014}=\frac{2013^2\cdot2014}{2013\cdot2014}=2013 $$
$$= \frac {(2013-2014)^3+2013^2\cdot 2014+1}{2013\cdot 2014}$$
$$=\frac {-1+2013^2\cdot 2014+1}{2013\cdot 2014}$$
$$=2013$$
Hint
Choose $2013=a$
$$\dfrac{a^3-2a^2(a+1)+3a(a+1)^2-(a+1)^3+1}{a(a+1)}$$
$$=\dfrac{-3a(a+1)+a(a+1)\{3(a+1)-2a\}}{a(a+1)}$$
$$=-3+3+a$$ as $a(a+1)\ne0$
HINT:
Let $a=2013$ and $b=2014$. You have:
$$\frac{a^3 - 2a^2b + 3ab^2 - b^3 + 1}{ab}$$
Recall that $(a-b)^3=a^3-3a^2b+3ab^2-b^3$ and use that to simplify.
