Answer
$x=-32$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
x^{7/5}=-128
,$ raise both sides to the power $\dfrac{5}{7}$ 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 $\dfrac{5}{7},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
\left( x^{\frac{7}{5}} \right)^{\frac{5}{7}}=(-128)^{\frac{5}{7}}
\\\\
x=(-128)^{\frac{5}{7}}
.\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[7]{-128}\right)^{5}
\\\\
x=\left(\sqrt[7]{(-2)^7}\right)^{5}
\\\\
x=\left( -2 \right)^{5}
\\\\
x=-32
.\end{array}
