Answer
$x=\{ -2,2 \}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x^{-6}=\dfrac{1}{64}
,$ 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}{x^6}=\dfrac{1}{64}
.\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(64)=x^6(1)
\\\\
64=x^6
\\\\
x^6=64
.\end{array}
Taking the $6th$ root of both sides, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[6]{x^6}=\pm\sqrt[6]{64}
\\\\
x=\pm\sqrt[6]{(2)^6}
\\\\
x=\pm2
.\end{array}
Hence, $
x=\{ -2,2 \}
.$
