Question

How do you simplify `((m^3p^5)/n^7)^6*((m^2n^0p^3)/(m^4n^2))^3` and write it using only positive exponents?

Answer 1

`(m^12color(white)(.)p^(39))/n^(48)`

Explanation:

`color(blue)("Preamble")`

Consider the context of `n^0`
`" "`suppose we had `2^3/2^3 =1`

`" "`This can be written as `2^(3-3) = 2^0=1`

`" "` so `n^0` disappears in the multiplication
....................................................................................
Consider the context of `((m^3p^5)/n^7)^6`

Suppose we had `p^2` then this is `pxxp`

So whatever `p` is it is multiplied by itself twice.

Set `p=3^2` then we have `3^2xx3^2 = 3xx3xx3xx3 = 3^4`

So `(3^2)^2=3^(2xx2)=3^4`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question - applying the above principles")`

Given:`" "((m^3p^5)/n^7)^6xx((m^2n^0p^3)/(m^4n^2))^3`

`=(m^(18)p^(30))/n^42 xx (m^6p^9)/(m^(12)n^6)`

`=(m^(18+6)color(white)(.)p^(30+9))/(m^12color(white)(.)n^(42+6))" "` this is the same as: `" "(m^12xxcancel(m^12))/(cancel(m^12)) xx(p^39)/(n^(48)`

`=(m^12color(white)(.)p^(39))/n^(48)`