Algebra

Algebraic Manipulation

Algebraic Manipulation: Level 3 Challenges

         

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

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

\[\large 2016^{x}+2016^{-x}=3\]

\[\large \sqrt{\frac{2016^{6x}-2016^{-6x}}{2016^{x}-2016^{-x}}} = \, ?\]

\[\Large\begin{align} \color{red}{x}\color{blue}{y} &= 7 \\ \color{red}{x}+\color{blue}{y} &= 5 \\ \color{red}{x}^3+\color{blue}{y}^3 &= \ \color{green}{?} \end{align}\]

\[\sum_{j=2}^{2016} \sum_{k=1}^{j-1} \dfrac kj = \frac{1}{2}+\frac{1}{3}+\frac{2}{3}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\cdots+\frac{2013}{2016}+\frac{2014}{2016}+\frac{2015}{2016}= \, ? \]

\[ \large {\begin{cases} x+y+z&=15 \\ xy+yz+xz&=72 \end{cases} } \]

Let \(x,y,\) and \(z\) be real numbers satisfying the system of equations above.

Find the possible range of \(x\).

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