Question

Why do you subtract exponents when dividing powers with the same base?

Answer 1

Because the factors in the numerator cancel with the factors in the denominator. Consider this example:

`(6^7)/(6^4) = (6*6*6*6*6*6*6)/(6*6*6*6) = 6*6*6 = 6^3 = 6^(7-4)`

All of the 6's in the denominator canceled with the 6's in the numerator, and only 3 6's "survived". The opposite can also happen, if there are more terms in the denominator:

`(2^2)/(2^4) = (2* 2)/(2* 2* 2 * 2) = (1)/(2^2) = 2^(-2) = 2^(2-4)`

So, everything is coherent