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## Algebraic Manipulation

Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.

# Level 2

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

Given that $$a$$, $$b$$ and $$c$$ are real numbers such that $$abc=1$$, find the value of $\left(\frac{a}{ab+a+1}+\frac{b}{bc+b+1}+\frac{c}{ca+c+1}\right)^2.$

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

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

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 $$x-y$$.

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