\(a=\dfrac{1+\sqrt{5}}{2};b=\dfrac{1-\sqrt{5}}{2}\Rightarrow\left\{{}\begin{matrix}a+b=1\\ab=\dfrac{\left(1+\sqrt{5}\right)\left(1-\sqrt{5}\right)}{4}=-1\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a^2+b^2=\left(a+b\right)^2-2ab=1^2+2=3\\a^3+b^3=\left(a+b\right)^3-3ab\left(a+b\right)=1+3=4\end{matrix}\right.\)
\(\Rightarrow a^5+b^5=\left(a^3+b^3\right)\left(a^2+b^2\right)-a^2b^2\left(a+b\right)=4.3-1.1=11\)
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