Funny you should give 27^7 as an example, because I actually did get good at powers of 2 and 3. 27^7 is 3^21 is 3×81^5, which is easy to calculate in your head if you're good at multiples and powers of 8: it's just binomial expansion with the next row after 1 4 6 4 1, i.e. 1 5 10 10 5 1. (I used to be able to directly recall 1 5 10 10 5 1 and 1 6 15 20 15 6 1, but this is literally the first time I'm doing non-trivial mental arithmetic in a decade.) Multiplying a power of 2 by 5 is the same as halving and multiplying by 10, which reduces the problem to a simple addition of digits of small powers of 2 (^0 = 1, ^2 = 4, ^6 = 64, ^9 = 512, ^11 = 2048, ^15 = 32768), then a multiplication by 3 – both of which are easy to perform in a streaming fashion, if you have a suitable delay line. (I use the auditory processing delay line ("echoic memory"), which would probably work better if I spoke a language like Mandarin, where all digits have the same syllable length – but I got by. Some find the mental abacus more reliable, but I have no training in this approach.)

I only memorised my powers of 2 up to 2^16, powers of 3 up to 3^5, and powers of 5 up to 5^5, because the part that made it fun was only memorising things I'd calculated myself, in my head, and this was only an occasional game. If my goal had been "develop the skill of quick arithmetic", I would've memorised the first 12 powers of every prime below 12, and my times tables up to 100×100 – but I resented times tables, so never really memorised them until I (briefly) got really into division.