Algebra
# Algebraic Manipulation

If $a-b=4$ and $ab=45$, what is the value of $a^3 -b^3$?

$\large \frac{1 + 3 + 5 + 7 +\ldots+ 199}{2 + 4 + 6 + 8 +\ldots+ 200} = \ ?$

$\large \frac{1^2+2^2+3^2+\cdots+2012^2+2013^2+2014^2}{1+2+3+\cdots+2012+2013+2014} = \ ?$

**Hint**: $1^2 + 2^2 + 3^2 + \cdots + n^2= \frac16 n(n+1)(2n+1)$.

$\large { \begin{cases} {x^2-y^2=4-2xy} \\ { x+ y = 2 } \end{cases} }$

If $x$ and $y$ satisfy the system of equations above, find the value of $x-y$.