Answer
$x=\left\{ -\dfrac{1}{4},\dfrac{1}{9}, \right\}$
Work Step by Step
Using $x^{-a}=\dfrac{1}{x^a}$, the given equation, $
x^{-2}-5x^{-1}-36=0
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{x^2}-\dfrac{5}{x}-36=0
.\end{array}
Multiplying both sides by the $LCD=
x^2
,$ then the solution to the given equation is
\begin{array}{l}\require{cancel}
1(1)-x(5)-x^2(36)=0
\\\\
1-5x-36x^2=0
\\\\
-36x^2-5x+1=0
\\\\
36x^2+5x-1=0
\\\\
(9x-1)(4x+1)=0
\\\\
x=\left\{ -\dfrac{1}{4},\dfrac{1}{9}, \right\}
.\end{array}
