Chapter 5.4, Problem 27E

To determine

To solve : the given equation.

Answer to Problem 27E

The solutions of the given equation are $x=8$ and $x=-8$ .

Explanation of Solution

Given information :

The given equation:

  $2x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=8$

Concept used :

Solving equations with rational exponent:

1. Isolate the power.
2. Raise each side of the equation to the reciprocal of the rotational exponent.

Calculation :

As per problem,

  $\begin{array}{l}2x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=8\text{ [Write original equation]}\\ x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=4\text{ [Divide each side by 2]}\\ $ {\text{(}{x}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.})}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=4^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\text{ [Raise each side to the power }\frac{3}{2}\text{ ]}$x=\pm \text{8 [Simplify]}\\ x=8\text{ or }x=-8\text{ [Write solutions separately]}\end{array}$

To check the solution, substitute the values of $x$ in the original equation

For $x=8$

  $\begin{array}{l}$ 2{(8)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\stackrel{?}{=}8\text{ [Substitute }x=8]$2(4)\stackrel{?}{=}8\text{ [Simplify}]\\ \text{ 8}=\text{8 }\end{array}$

For $x=-8$

  $\begin{array}{l}$ 2{(-8)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\stackrel{?}{=}8\text{ [Substitute }x=-8]$2(4)\stackrel{?}{=}8\text{ [Simplify}]\\ \text{ 8}=\text{8 }\end{array}$

Hence, the solutions of the equation are $x=8$ and $x=-8$ .