jika [tex] a^{2x} = \sqrt{2} +1[/tex], maka [tex] \frac{ a^{3x}+ {a^ {-3x} } }{ a^{x}+a^{-x }} [/tex] = ??

[tex]$\begin{align} \display \bold{Diketahui :~~ a^{2x} = \sqrt{2} + 1} \end{align}[/tex]

[tex]$\begin{align} \display \frac{a^{3x} + a^{-3x}}{a^x + a^{-x}} &= \frac{a^x(a^{2x } + a^{-4x})}{a^x(1 + a^{-2x})} \\ &= \frac{a^{2x} + a^{-4x}}{1 + a^{-2x}} \\ &= \frac{a^{2x} + (a^{2x})^{-2}}{1 + (a^{2x})^{-1}} \\ &= \frac{( \sqrt{2} + 1 ) + ( \sqrt{2} + 1)^{-2}}{1 + ( \sqrt{2} + 1)^{-2}} \\ \bold{misal: ~ ~ b = ( \sqrt{2}  + 1)} \\ &= \frac{b + b^{-2}}{1 + b^{-2}} \\ &= \frac{b + \frac{1}{b} }{1 + \frac{1}{b} } \\ &= \frac{b^3 + 1}{b^2 + 1} \\ &= \frac{( \sqrt{2} + 1)^3 + 1}{(\sqrt{2} + 1)^{2} + 1} \end{align}[/tex]