## Actually, 17 bytes (non-competing)

    ;a¡;lrR(♀ⁿ@♂≈♀*;░

[Try it online!](http://actually.tryitonline.net/#code=O2HCoTtsclIo4pmA4oG_QOKZguKJiOKZgCo74paR&input=MTIzNDU2CjEw)

This submission is non-competing because the `♀` command was added after this challenge.

Explanation:

    ;a¡;lrR(♀ⁿ@♂≈♀*;░
                       initial stack: [b n] (b = base, n = number)
    ;                  dupe b
     a                 invert stack
      ¡                n as a base-b integer
       ;lrR            dupe, length, range, reverse
           (♀ⁿ         raise b to each power in range
              @♂≈      create list of integers from base-b string
                 ♀*    pairwise multiplication
                   ;░  filter out zeroes