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# Is there a 'distributive property of division'?

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Is there a "distributive property of division"??

Like as in this question:

(-3)2n+1 /(27*(-3)2n)  n is a positive whole number.

Can you do this:

((-3)2n+1 /27)*((-3)2n+1 /(-3)2n) ??????

THANK YOU

Aug 26, 2017

#1
+102320
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(-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    am / an  = a ( m - n)

So   ...we have....

(-3) [ (2n + 1) - 2n ]  *  (1/27)  =

(-3)1  * (1/27)  =

(-3)  / 27  =

-1 / 9

Aug 26, 2017
#2
+8577
+4

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....

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

Aug 26, 2017
edited by hectictar  Aug 27, 2017