Is there a 'distributive property of division'?

(-3⁾²ⁿ⁺¹ /(27*(-3)²ⁿ)

Note that we can write this as

(-3⁾²ⁿ⁺¹ / (-3)²ⁿ   * ( 1 /27 )

And remember that we have the property that    a^m / aⁿ  = a ^{( m - n)}

So   ...we have....

(-3) ^{[ (2n + 1) - 2n ]}  *  (1/27)  =

(-3)¹  * (1/27)  =

(-3)  / 27  =

-1 / 9

Also...

\(\frac{(-3)^{2n+1}}{27\,\cdot\,(-3)^{2n}}=\frac{(-3)^{2n+1}}{27}\,\cdot\,\frac{(-3)^{2n+1}}{(-3)^{2n}}\)

This is not true.

If you multiply the two fractions on the right side together, you will get   \(\frac{[ (-3)^{2n+1})]^2}{27\,\cdot\,(-3)^{2n}}\)     .

\(\frac{a}{bc}\,\neq\,\frac{a}{b}\,\cdot\,\frac{a}{c}\)

But..you can distribute division the same as you distribute multiplication, like this....

\(\frac{8 + 6 +10}{2}=\frac12(8+6+10)\,=\,(\frac12)(8)+(\frac12)(6)+(\frac12)(10)\,=\,4+3+5\,=\,12\)