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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 \(|ad + bc|\).

If \(x + \dfrac{1}{x} = 3\), then what is \(x^5 + \dfrac{1}{ x^5 } \)?

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

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

\[S = \displaystyle\sqrt{9 - \sqrt{\dfrac{13}{9} + \sqrt{\dfrac{13}{81} - \sqrt{\dfrac{13}{6561} + \sqrt{\dfrac{13}{43046721} - .......}}}}}\]

If \(S = \sqrt{\dfrac{a}{b}}\) where \(a, b\) are both primes, find \(a + b\).

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