Answer
$\left\{-\dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{3}}{2}\right\}$
Work Step by Step
Divide 64 on both sides of the equation to obtain:
$x^6=\dfrac{27}{64}$
Write $64$ as $2^6$ and 27 as $3^3$ to obtain:
$x^6=\dfrac{3^3}{2^6}$
Take the sixth root of both sides to obtain:
$x = \pm \sqrt[6]{\dfrac{3^3}{2^6}}
\\x-\pm \dfrac{\sqrt[6]{3^3}}{2}$
Use the rule $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ to obtain:
$x=\pm \dfrac{3^{\frac{3}{6}}}{2}$
Simplify the rational exponent to obtain:
$x = \pm \dfrac{3^{\frac{1}{2}}}{2}$
Use the rule $a^{\frac{1}{2}}=\sqrt{a}$ to obtain:
$x= \pm \dfrac{\sqrt{3}}{2}$
Thus, the solution set of the given equation is $\left\{-\dfrac{\sqrt{3}}{2}, \dfrac{\sqrt{3}}{2}\right\}$.
