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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 3

\[\Large\begin{align} \color{red}{x}\color{blue}{y} &= 7 \\ \color{red}{x}+\color{blue}{y} &= 5 \\ \color{red}{x}^3+\color{blue}{y}^3 &= \ \color{green}{?} \end{align}\]

\[ \large x^{63} + x^{44} + x^{37} + x^{31} + x^{26} + x^9 + 6 \]

If \(x\) satisfies the equation \( \left(x + \dfrac1x\right)^2 = 3 \), then find the value of the expression above.

Given that \(a^2 + b^2 = c^2 + d^2 = 85\) and \(|ac - bd| = 40,\) find \(|ad + bc|\).

\[\large S = 2014^{3}-2013^{3}+2012^{3}-2011^{3}+\cdots+2^{3}-1^{3}\]

What is the largest perfect square that divides \(S\) above?

\[\large 2016^{x}+2016^{-x}=3\]

\[\large \sqrt{\frac{2016^{6x}-2016^{-6x}}{2016^{x}-2016^{-x}}} = \, ?\]

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