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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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