Solve y=k^4+sqrt{t}t

$$y = k ^ { 4 } + \sqrt { t } t$$

Solve for k (complex solution)

$$k=i\sqrt[4]{y-t^{\frac{3}{2}}}$$

$$k=\sqrt[4]{y-t^{\frac{3}{2}}}$$

$$k=-\sqrt[4]{y-t^{\frac{3}{2}}}$$

$$k=-i\sqrt[4]{y-t^{\frac{3}{2}}}$$

Solve for t (complex solution)

$$t=\left(y-k^{4}\right)^{2}$$

$$y=k^{4}\text{ or }arg(y-k^{4})<\pi$$

Solve for k

$$\left\{\begin{matrix}k=\sqrt{-\sqrt{y-t^{\frac{3}{2}}}}\text{; }k=-\sqrt{-\sqrt{y-t^{\frac{3}{2}}}}\text{, }&t\geq 0\text{ and }y=t^{\frac{3}{2}}\\k=\sqrt[4]{y-t^{\frac{3}{2}}}\text{; }k=-\sqrt[4]{y-t^{\frac{3}{2}}}\text{, }&y\geq t^{\frac{3}{2}}\text{ and }t\geq 0\end{matrix}\right.$$

Solve for t

$$t=\left(y-k^{4}\right)^{\frac{2}{3}}$$

$$y-k^{4}\geq 0$$