Solve for a^6
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Solution
\[\begin{gathered}
a^2-a-1=0 ,a>0\\\\
a=\frac{-(-1) \pm \sqrt{(-1)^2-4(1)(-1)}}{2(1)} \\\\
a=\frac{1+\sqrt{5}}{2}, a=\frac{1-\sqrt{5}}{2}<0 \\\\
\Rightarrow a=\frac{1+\sqrt{5}}{2}
\end{gathered}\]
\[\begin{aligned}
&\left(a^2\right)^3=(a+1)^3 \\\\
&a^6=a^3+3 a^2(1)+3 a(1)^2+(1)^3 \\\\
&a^6=a^3+3 a^2+3 a+1 \\\\
&a^6=a^2+a+3 a^2+3 a+1 \\\\
&a^6=4 a^2+4 a+1 \\\\
&a^6=4(a+1)+4 a+1 \\\\
&a^6=4 a+4+4 a+1 \\\\
&a^6=8 a+5
\end{aligned}\]
\[\begin{aligned}
a^6 &=8 a+5 \\\\
a^6 &=8\left(\frac{1+\sqrt{5}}{2}\right)+5 \\\\
a^6 &=4(1+\sqrt{5})+5 \\\\
a^6 &=4+4 \sqrt{5}+5 \\\\
a^6 &=9+4 \sqrt{5}
\end{aligned}\]
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