Chapter 5.4, Problem 36E

To determine

To correct : the error in solving the given equation.

Answer to Problem 36E

In the incorrect solution, $8$ on the left-hand side is not raised to the power two-third.

The correct solution of the given equation is $x=25$ .

Explanation of Solution

Given information :

The given equation:

  $8x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=1000$

The incorrect solution of the equation:

  $\begin{array}{l}8x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=1000\\ 8{\left(x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}={1000}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\\ 8x=100\\ x=\frac{25}{2}\end{array}$

Concept used :

Solving equations with rational exponent:

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

Calculation :

As per problem,

Consider the incorrect solution,

  $\begin{array}{l}8x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=1000\\ 8{\left(x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}={1000}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\\ 8x=100\\ x=\frac{25}{2}\end{array}$

While raising the each side to the power two-third, $8$ on the left-hand side is not raised to the power two-third.

Correcting the error and solving the equation:

  $\begin{array}{l}8x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=1000\text{ [Write original equation]}\\ {\left(8x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}={1000}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\text{ [Raise each side to the power}\frac{2}{3}\text{]}\\ \text{ 4}x=100\text{ [Simplify]}\\ x=25\text{ [Divide each side by 4]}\end{array}$

Therefore, the correct solution of the equation is $x=25$ .