Chapter 8.2, Problem 64STP

To determine

To solve: The equation ${\left(3x-2\right)}^{\frac{1}{5}}+6=5$ .

Answer to Problem 64STP

The solution of the equation ${\left(3x-2\right)}^{\frac{1}{5}}+6=5$ is $x=\frac{1}{3}$ .

Explanation of Solution

Given:

The equation is ${\left(3x-2\right)}^{\frac{1}{5}}+6=5$ .

Calculation:

Simplify the equation ${\left(3x-2\right)}^{\frac{1}{5}}+6=5$ .

  $\begin{array}{c}{\left(3x-2\right)}^{\frac{1}{5}}=5-6\\ {\left(3x-2\right)}^{\frac{1}{5}}=-1\end{array}$

Rising to the power of $5$ in the equation ${\left(3x-2\right)}^{\frac{1}{5}}=-1$ .

  $\begin{array}{c}{\left({\left(3x-2\right)}^{\frac{1}{5}}\right)}^5={\left(-1\right)}^5\\ 3x-2=-1\\ 3x=-1+2\\ 3x=1\end{array}$

Simplify further.

  $x=\frac{1}{3}$

Therefore, the solution of the equation ${\left(3x-2\right)}^{\frac{1}{5}}+6=5$ is $x=\frac{1}{3}$ .