Is it a Satisfying Number? inspired by this chat conversation

A satisfying number is a number whose decimal representation is of the form abx, with the following properties:


x is the longest trailing repeating suffix, or the last digit if there is no repetition at the end (123333 -> 3333, 545656 -> 5656, 123 -> 3)
b is the single digit prior to x (123333 -> 2, 55545656 -> 4)
a is the remaining prefix (123333 -> 1, 55545656 -> 555)
a == c**b (** denotes exponentation), where c is the number of repetitions of the smallest repeating portion of x (1623333 -> 4 (3 3 3 3, not 33 33))


For example, 8300 is a satisfying number with a = 8, b = 3, c = 2, and x = 00. 24651 is not a satisfying number, because x = 1, b = 5, a = 246, and there is no integer c that satisfies c^5 = 246. 1222 is also not a satisfying number, because with x = 222 and b = 1, there are no remaining digits for a.

Given a positive integer n &gt;= 100, output whether or not n is a satisfying number.

Examples

8300: True (a=8, b=3, c=2, x=00)
24651: False 
1222: False
92555: True (a=9, b=2, c=3, x=555)
64633: True (a=64, b=6, c=2, x=33)
512944: True (a=512, b=9, c=2, x=44)
123: True (a=1, b=2, c=1, x=3)
822809: False 
376664: False 
723799: False 
1234: False 
34330000000: True (a=343, b=3, c=7, x=0000000)
92313131: True (a=9, b=2, c=3, x=313131)
16424442444: True (a=16, b=4, c=2, x=24442444)
Jelly, 26 bytes

feels too long

DŒṖṖEÐƤḄ$ÐṀLÐṂṪµḢ*@0¦LµṪ⁼Ḍ


A monadic link taking an integer and returning 1 if the input is satisfying and 0 if not.

Try it online! or see a test-suite

How?

DŒṖṖEÐƤḄ$ÐṀLÐṂṪµḢ*@0¦LµṪ⁼Ḍ - Link: integer, n    e.g. 8300
D                          - to decimal list          [8,3,0,0]
 ŒṖ                        - all partitions           [[[8],[3],[0],[0]],[[8],[3],[0,0]],[[8],[3,0],[0]],[[8],[3,0,0]],[[8,3],[0],[0]],[[8,3],[0,0]],[[8,3,0],[0]],[[8,3,0,0]]]
   Ṗ                       - pop (the "no tail" one)  [[[8],[3],[0],[0]],[[8],[3],[0,0]],[[8],[3,0],[0]],[[8],[3,0,0]],[[8,3],[0],[0]],[[8,3],[0,0]],[[8,3,0],[0]]]
         ÐṀ                - keep maximal under the operation: e.g. for [[8,3],[0],[0]]
        $                  -   last two links as a monad:
     ÐƤ                    -     for suffixes:   i.e. [[[8,3],[0],[0]],[[0],[0]],[[0]]]
    E                      -       all equal?         [0              ,1        ,1]
       Ḅ                   -     convert from binary  3
                           -          ...which yields [[[8],[3],[0],[0]],[[8,3],[0],[0]]]
            ÐṂ             - keep minimal under the operation:
           L               -   length
                           -          ...which yields [[[8,3],[0],[0]]]
              Ṫ            - tail (rightmost)         [[8,3],[0],[0]] 
               µ           - monadic chain separation
                Ḣ          - yield head and modify    [8,3]   ...leaving [[0],[0]]
                     L     - length (of modified)     2
                    ¦      - sparse application       (to the [8,3])
                   0       -   ...to index: 0         (to the rightmost digit, the 3)
                 *@        -   power (sw@p args)      [8,8]  ([8, 3*@2] = [8, 2*3] = [8,8])
                      µ    - monadic chain separation
                       Ṫ   - yield tail and modify    8   ...leaving [8]
                         Ḍ - from decimal (modified)  8
                        ⁼  - equal?                   1