Answer
$x=-3$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\left( \sqrt[3]{5}\right)^{-x}=\left(\dfrac{1}{5}\right)^{x+2}
,$ express both sides of the equation in the same base. Once the bases are the same, equate the exponents. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
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}
\left( 5^{\frac{1}{3}} \right)^{-x}=\left(\dfrac{1}{5}\right)^{x+2}
.\end{array}
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}
\left( 5^{\frac{1}{3}} \right)^{-x}=\left( 5^{-1}\right)^{x+2}
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
5^{\frac{1}{3}(-x)}=5^{-1(x+2)}
\\\\
5^{-\frac{1}{3}x}=5^{-x-2}
.\end{array}
Since the bases are the same, then the exponents can be equated. Hence,
\begin{array}{l}\require{cancel}
-\frac{1}{3}x=-x-2
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
3\left( -\frac{1}{3}x \right)=(-x-2)3
\\\\
-x=-3x-6
\\\\
-x+3x=-6
\\\\
2x=-6
\\\\
x=-\dfrac{6}{2}
\\\\
x=-3
.\end{array}
