#### Answer

$x=3$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\dfrac{1}{27}=x^{-3}
,$ use the laws of exponents and the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{27}=\dfrac{1}{x^3}
.\end{array}
Since $\dfrac{a}{b}=\dfrac{c}{d}$ implies $ad=bc$ or sometimes referred to as cross-multiplication, the equation above is equivalent to
\begin{array}{l}\require{cancel}
1(x^3)=1(27)
\\\\
x^3=27
.\end{array}
Taking the cube root of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{x^3}=\sqrt[3]{27}
\\\\
x=\sqrt[3]{(3)^3}
\\\\
x=3
.\end{array}