How to rewrite this problem in an equivalent SOCP form?

Question

Suppose I have problem like this: \begin{equation*} \min_x\left\{f^Tx:\sum^n_{i=1}\left\lvert x_i-x^0_i\right\rvert^{3/2} \le t\right\}, \end{equation*} How can I recast it into SOCP? I have some useful lemma such as $z^{3/2}\le t \iff \exists w: wt\ge z^2, z\ge w^2$, and $wt\ge z^2\iff \left\lVert \begin{array}{c}2z\\w-t\end{array}\right\rVert_2 \le w+t$.