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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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