180° about z takes (x,y,z) -> (-x,-y,z)

90° about x takes (x,y,z) -> (x,z,-y)
so, your two rotations take (x,y,z) -> (-x,z,y)... z axis top/bottom, x/y axis right/ left... A dice has 3 on the top, 4 bottom, 1 left, 2 right, 5 left back, 6 right back... after the above rotations what numbers are showing on the dice now?

To determine which numbers are showing on the dice after the rotations, we can start with the initial positions of the numbers and apply the given rotations.

The initial positions of the numbers on a standard dice are:
- Top: 3
- Bottom: 4
- Left: 1
- Right: 2
- Front: 5
- Back: 6

Using the given rotations:

1. The rotation of 180° about the z-axis takes (x,y,z) to (-x,-y,z).
The top becomes (-3), the bottom becomes (-4), the left becomes (-1), the right becomes (-2), the front remains the same (5), and the back remains the same (6).

2. The rotation of 90° about the x-axis takes (x,y,z) to (x,z,-y).
Applying this rotation to the updated positions after the first rotation:
- The top remains the same (-3)
- The bottom remains the same (-4)
- The left becomes (5)
- The right becomes (-2)
- The front remains the same (6)
- The back becomes (-1)

Therefore, after applying both rotations, the numbers showing on the dice are:
- Top: -3
- Bottom: -4
- Left: 5
- Right: -2
- Front: 6
- Back: -1