Curve in an algebraic variety

Question

Let $\lambda_1, \lambda_3, \lambda_3$ be distinct real numbers. Can it be that a curve of the form $$ t \mapsto \gamma(t) := (e^{\lambda_1 t}, e^{\lambda_2 t}, e^{\lambda_3 t}) $$ is contained for all times $t \ge 0$ in an algebraic variety of the form $$ a_1 x_1^k + a_2 x_2^k + a_3 x_3^k = 0 ? $$

Answer 1

No. Then we would have $a_1 e^{\lambda_1 k t} + a_2 e^{\lambda_2 k t} + a_3 e^{\lambda_3 k t}$ for all $t$. But the $e^{\lambda_i t}$ are linearly independent as functions (which follows from, say, the Vandermonde determinant, though for real $\lambda_i$ you can show this with a limit argument), so $a_1=a_2=a_3=0$.