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:

$\dfrac{6^7}{6^4} = \dfrac{6\times6\times6\times6\times6\times6\times6}{6\times6\times6\times6} = 6\times6\times6 = 6^3 = 6^{7-4}$

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:

$\dfrac{2^2}{2^4} = \dfrac{2\times 2}{2\times 2\times 2 \times 2} = \dfrac{1}{2^2} = 2^{-2} = 2^{2-4}$

So, everything is coherent