Answer
$x=\{ -4,4 \}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $ x^{-4}=\dfrac{1}{256} ,$ 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^4}=\dfrac{1}{256} .\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(256)=x^4(1) \\\\ 256=x^4 \\\\ x^4=256 .\end{array}
Taking the $4th$ root of both sides, the equation above is equivalent to \begin{array}{l}\require{cancel} \sqrt[4]{x^4}=\pm\sqrt[4]{256} \\\\ x=\pm\sqrt[4]{(4)^4} \\\\ x=\pm4 .\end{array}
Hence, $
x=\{ -4,4 \}
.$
