Negative indices with fraction, why change the sign when swapping

Question

Given $\dfrac{a^{-6t}b^{6t}}{a^{3t}b^{t}}$, the solution is $\dfrac{b^{5t}}{a^{9t}}$. That is done by swapping the denominator and numerator and by changing their signs to the opposite by using this index law $a^{-n} = \dfrac{1}{a^n}$ but I don't understand how exactly the swapping is done, when I do swap the I usually just multiply or divide by a factor in either the denominator or numerator to essentially achieve the swap, which is also what I assumed is to be done here, but even with the index law I don't see why it would make sense to change the sign of the exponent. $$ \frac{a^{-6t}b^{6t}}{a^{3t}b^{t}} = \frac{\frac{1}{a^{6t}}b^{6t}}{a^{3t}b^t} = \frac{\frac{b^{6t}}{a^{6t}} }{a^{3t} b^t} = \frac{b^{6t}}{a^{6t}} / {a^{3t}b^t} = \frac{b^{6t}}{a^{6t}} \cdot \frac{1} {a^{3t}b^t} = \frac{b^{6t}}{a^{9t}b^t} $$ This is quite different from how its supposed to be done, but I hope this may help on what I am doing.

Answer 1

In higher math we get rid of division and instead multiply by reciprocals. The advantage is that for multiplication is associativity, which is that $(ab)c=a(bc)$ but for division $a/(b/c) \neq (a/b)/c$. So this means when we write $a/b$ what we really mean is $ab^{-1}$ to ensure the operation is multiplication. Looking at your expression we have $$\frac {a^{-6t}b^{6t}}{a^{3t}b^{t}} = {a^{-6t}b^{6t}}(a^{3t}b^t)^{-1} = a^{-6t}b^{6t}a^{-3t}b^{-t}=a^{-9t}b^{5t}=\frac{b^{5t}}{a^{9t}}$$ and we have the result.