Let us consider the solutions to the equation in the complex number $z$
$$z ^ { 4 } = - 324 \quad \cdots (1)$$
and the solutions to the equation in the complex number $z$
$$z ^ { 4 } = t ^ { 4 } \quad \cdots (2)$$
where $t$ is a positive real number.
(1) To find the solutions to (1), let us set
$$z = r ( \cos \theta + i \sin \theta ) \quad ( r > 0,0 < \theta \leqq 2 \pi )$$
Then
$$z ^ { 4 } = r ^ { \mathbf { M } } ( \cos \mathbf { N } \theta + i \sin \mathbf { N } \theta ) .$$
The values of $r$ and $\theta$ such that this expression is equal to $-324$ are
$$\begin{aligned} & r = \mathbf { O } \sqrt { \mathbf { P } } , \\ & \theta = \frac { \mathbf { Q } } { \mathbf { R } } \pi , \frac { \mathbf { S } } { \mathbf { R } } \pi , \frac { \mathbf { T } } { \mathbf { R } } \pi , \frac { \mathbf { U } } { \mathbf { R } } \pi , \end{aligned}$$
where $\mathbf { Q } < \mathbf { S } < \mathbf { T } < \mathbf { U }$.
(2) There are $\mathbf { V }$ solutions to equation (2), and these solutions are dependent on $t$. Now, consider any one of the solutions to (1) and any one of the solutions to (2), and let $d$ be the distance on the complex number plane between these two solutions. Then, over the interval $0 < t \leqq 4$, the minimum value of $d$ is $\mathbf { W }$ and the maximum value is $\sqrt { \mathbf { X Y } }$.