We are to find the positive number $a$ satisfying $a ^ { 3 } = 9 + \sqrt { 80 }$. Let us consider the positive number $b$ which satisfies $b ^ { 3 } = 9 - \sqrt { 80 }$. Then $$\left\{ \begin{aligned}
a ^ { 3 } + b ^ { 3 } & = \mathbf { A B } \\
a b & = \mathbf { C }
\end{aligned} \right.$$ holds. First, using (2), (1) can be transformed into $$( a + b ) ^ { 3 } - \mathbf { D } ( a + b ) = \mathbf { A B } .$$ Then, setting $x = a + b$, we have $$x ^ { 3 } - \mathrm { D } x = \mathrm { AB } .$$ Transforming this equation, we obtain $$x ^ { 3 } - 27 = \mathbf { D } ( x - \mathbf { E } ) \text {, }$$ which gives $$( x - \mathbf { F } ) \left( x ^ { 2 } + \mathbf { G } x + \mathbf{H} \right) = 0 .$$ From that we have $x = \mathbf{I}$ and hence $$a + b = \mathbf{I} \text {. }$$ Thus, from (2), (3) and $a > b$, we have $$a = \frac { \mathbf { J } + \sqrt { \mathbf { K } } } { \mathbf{L} } .$$
We are to find the positive number $a$ satisfying $a ^ { 3 } = 9 + \sqrt { 80 }$.
Let us consider the positive number $b$ which satisfies $b ^ { 3 } = 9 - \sqrt { 80 }$.\\
Then
$$\left\{ \begin{aligned}
a ^ { 3 } + b ^ { 3 } & = \mathbf { A B } \\
a b & = \mathbf { C }
\end{aligned} \right.$$
holds.
First, using (2), (1) can be transformed into
$$( a + b ) ^ { 3 } - \mathbf { D } ( a + b ) = \mathbf { A B } .$$
Then, setting $x = a + b$, we have
$$x ^ { 3 } - \mathrm { D } x = \mathrm { AB } .$$
Transforming this equation, we obtain
$$x ^ { 3 } - 27 = \mathbf { D } ( x - \mathbf { E } ) \text {, }$$
which gives
$$( x - \mathbf { F } ) \left( x ^ { 2 } + \mathbf { G } x + \mathbf{H} \right) = 0 .$$
From that we have $x = \mathbf{I}$ and hence
$$a + b = \mathbf{I} \text {. }$$
Thus, from (2), (3) and $a > b$, we have
$$a = \frac { \mathbf { J } + \sqrt { \mathbf { K } } } { \mathbf{L} } .$$