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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.
Given that \(a^2 + b^2 = c^2 + d^2 = 85\) and \(|ac - bd| = 40,\) find \(|ad + bc|\).
by
Alan Yan
If \(x + \dfrac{1}{x} = 3\), then what is \(x^5 + \dfrac{1}{ x^5 } \)?
by
yaman malla
\[\Large\begin{align} \color{red}{x}\color{blue}{y} &= 7 \\ \color{red}{x}+\color{blue}{y} &= 5 \\ \color{red}{x}^3+\color{blue}{y}^3 &= \ \color{green}{?} \end{align}\]
by
king james Palma
\[ \large x^{63} + x^{44} + x^{37} + x^{31} + x^{26} + x^9 + 6 \]
If \(x\) satisfies the equation \( \left(x + \dfrac1x\right)^2 = 3 \), then find the value of the expression above.
by
Sanjeet Raria
\[S = \displaystyle\sqrt{9 - \sqrt{\dfrac{13}{9} + \sqrt{\dfrac{13}{81} - \sqrt{\dfrac{13}{6561} + \sqrt{\dfrac{13}{43046721} - .......}}}}}\]
If \(S = \sqrt{\dfrac{a}{b}}\) where \(a, b\) are both primes, find \(a + b\).
by
Brian Charlesworth