If a² + a + 1 = 0, then the value of a⁹ is?

$${a^2} + a + 1 = 0$$
\[\left[ \begin{array}{l} {a^3} + {1^3} = \left( {a + 1} \right)\left( {{a^2} + a + 1} \right)\\ {a^3} - {1^3} = \left( {a - 1} \right)\left( {{a^2} + a + 1} \right) \end{array} \right]\]
$$\eqalign{ & \therefore \left( {{a^3} - 1} \right) = \left( {a - 1} \right) \times 0 \cr & \Rightarrow {a^3} - 1 = 0 \cr & \Rightarrow {a^3} = 1 \cr & \Rightarrow {\left( {{a^3}} \right)^3} = {1^3} \cr & \Rightarrow {a^9} = 1 \cr} $$