# Chapter 4 - Section 4.2 - Exponential Functions - 4.2 Exercises: 86

$x=-32$

#### 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}

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