Identity for exponential character sums

Question

I was confused about the following identity I ran into. I would appreciate it if somebody could clear this up for me. Suppose that we have an exponential sum $$g(a)=\sum_{t=0}^{p-1} \exp( 2\pi i at^k/p),$$ where $p$ is prime and $k\mid p-1$. Then one can write $$g(a)=\sum_{\chi^k=\chi_0, \chi\neq \chi_0}\overline{\chi}(a) \sum_{t=0}^{p-1}\chi(t)\exp(2\pi i at/p).$$ Here, the sum is taken over all non-principal multiplicative characters of $\mathbb{F}_p^{\times}$ such that $\chi^k$ is the principal character. I feel that somehow orthongoality realtions need to be used, but I'm not sure what to try. Thanks!