If α and β are the zeroes of $x^2-3x+1$, the value of α⁴ + β⁴ is:

Build up in two steps. First ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta =9-2=7$. Then ${\alpha }^4+{\beta }^4=({\alpha }^2+{\beta }^2{)}^2-2(\alpha \beta {)}^2=49-2(1{)}^2=47$. 49 stops at $({\alpha }^2+{\beta }^2{)}^2$; 81 raises the sum straight to the fourth power, $3^4$; 45 subtracts $4\alpha \beta$ instead of $2(\alpha \beta {)}^2$.