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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 1

$\large{\color{blue}x = \sqrt\frac{\color{red}y - 1}{\color{red}y +1}}$
Express $$y$$ in terms of $$x$$.

$\Large\sqrt[3]{3\sqrt[3]{3\sqrt[3]{3\cdots}}}=\sqrt{n} \quad, \quad n = \ ?$

$$x=\sqrt { 2+\sqrt { 2+\sqrt { 2+\sqrt { 2\ldots } } } }$$

What is the value of $$x$$?

$\huge{ x^{x^{x^{x^{x^{.^{.^.}}}}}}=2}$

Find the value of $$x$$ that satisfies the equation above.

$\large \sqrt{12+\sqrt{12+\sqrt{12+\sqrt{12 + ...}}}}= \ ?$

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