I don't know if this matrix has a name, but I want to prove the determinant is not equal to zero: $$ \begin{vmatrix} 1 & 1 & \cdots & 1\\ 1 & 2^1 & \cdots & n^1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 2^{n-1} & \cdots & n^{n-1} \end{vmatrix} \neq 0 $$
I know the formula for the determinant is $\sum_P\epsilon_P a_{1,j_1}\cdots a_{n,j_n}$, but I don't see any way to use it or another to prove the above statement.
Background: I want to prove this in order to prove that functions $e^x,e^{2x},..,e^{nx}$ are independent, by using the determinant of the Wronskian and then taking out the $e^x,..,e^{nx}$ from each column, where the above matrix remains. My math skills are not advanced enough to use much more than the Wronskian I am afraid for this task.
Can someone help me? Thanks in advance.
