## College Algebra (11th Edition)

$x=\{ -2,2 \}$
$\bf{\text{Solution Outline:}}$ 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, $x=\{ -2,2 \} .$