A carton can hold 1'000 unit cubes that measure 1 inch by 1inch by 1 inch. How can describe the dimensions of the carton using unit cubes

You can't tell the dimensions from that information.

All you can say is that the length multiplied by the width,

and then that product multiplied by the height, is 1,000.

But you can't say what the length or the width or the height is.

A few possibilities:

Length Width Height

1 1 1000

2 1 500

4 5 50

5 8 25

10 10 10

10 20 5

There are many more possibilities.