## Solve for a^6

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### Solution

\[\begin{aligned}
&\left(a^2\right)^3=(a+1)^3 \\\\
&a^6=a^3+3 a^2(1)+3 a(1)^2+(1)^3 \\\\
&a^6=a^3+3 a^2+3 a+1 \\\\
&a^6=a^2+a+3 a^2+3 a+1 \\\\
&a^6=4 a^2+4 a+1 \\\\
&a^6=4(a+1)+4 a+1 \\\\
&a^6=4 a+4+4 a+1 \\\\
&a^6=8 a+5
\end{aligned}\]

\[\begin{aligned}
a^6 &=8 a+5 \\\\
a^6 &=8\left(\frac{1+\sqrt{5}}{2}\right)+5 \\\\
a^6 &=4(1+\sqrt{5})+5 \\\\
a^6 &=4+4 \sqrt{5}+5 \\\\
a^6 &=9+4 \sqrt{5}
\end{aligned}\]

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