Square-Cube Digit Usage
05AB1E, 14 12 13 bytes
-2 bytes thanks to @KevinCruijssen
+1 byte thanks to @Grimmy
∞.Δ23SmJœIÅ?Z

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Explanation
∞.Δ                  - First number that... 
   23Sm              - Power of 2 and 3 [n^2, n^3] 
       J             - Concatenated
        œ            - Permutations of this number
         IÅ?Z        - any of these start with the number 

By the way it's quite slow...