#### Answer

$x=-27$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x^{5/3}=-243
,$ raise both sides to the power $\dfrac{3}{5}$ to make the exponent of the variable equal to $1.$ Then use the definition of rational exponents and the concepts of radicals to solve for the variable.
$\bf{\text{Solution Details:}}$
Raising both sides of the equation to the power $\dfrac{3}{5},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
\left( x^{\frac{5}{3}} \right)^{\frac{3}{5}}=(-243)^{\frac{3}{5}}
\\\\
x=(-243)^{\frac{3}{5}}
.\end{array}
Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x=\left(\sqrt[5]{-243}\right)^{3}
\\\\
x=\left(\sqrt[5]{(-3)^5}\right)^{3}
\\\\
x=\left( -3 \right)^{3}
\\\\
x=-27
.\end{array}