How to do exponential division when base and exponent is different between numerator and denominator?

Question

How to do division when

(a power x) / (b power y)

Say, a,x,b,y are distinct values.

For example,

(2 power 19) / (7 power 23). Doing mathematical operations with bigger exponents is tedious and error-prone.

Is there any way to simplify above expression such that the final decimal result wouldn't change?

Is there any way to simplify this such that after simplifying, I don't have to calculate exponent > 10, say.

I had the same question posted in sof : 

https://stackoverflow.com/questions/58717012/how-to-do-exponential-division-when-base-and-exponent-is-different-between-numer

But the problem with that solution is I'm ending up calculating huge exponents even after simplifying.

Answer 1

Well, you could use the same trick, going calculation to the base you would like to see the answer. I assume base 10 for simplicity

$$ 2^{19 - 23\log_2{7}} = 10^{\log_{10}2\;(19 - 23\log_2{7})}=10^{-13.717685}=10^{0.282315}\;10^{-14}= 1.915645\;10^{-14} $$

You don't have to compute huge exponent ($10^{-14}$), it goes directly to output