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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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