Question

How do you solve `x^ (-2/3) = 9`?

Answer 1

raise both sides to the `-3/2` power.

Then you have:
`x^((-2/3)(-3/2))=9^(-3/2)`

`x=9^(-3/2)=(3^2)^(-3/2)`

`x= 3^-3`

`x=1/27`

Answer 2

A trick using logs.

`x=1/27`

Explanation:

Given: `x^(-2/3)=9`

Take logs of both sides:

`ln(x^(-2/3))`=ln(9)#

`-2/3ln(x)=ln(9)`

`ln(x)=-3/2ln(9)`

`x=ln^(-1)[-3/2ln(9)]`

`x= 0.037color(white)(.)bar(037)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`1000x=37.037bar(037)`
`ul(color(white)(1000)x=color(white)(0)0.037bar(037)larr" Subtract")`
`color(white)("d") 999x=37`

`x=37/999 = 1/27`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Check")`

`x^(-2/3) = 1/(x^(2/3))`

Consider `x^(2/3)->(1/27)^(2/3)= 0.11bar1`

`1/(x^(2/3))=1/(0.11bar1) = 9 color(red)(larr" As required")`