College Algebra, Chapter 1, 1.1, Section 1.1, Problem 74

The equation $6x^{\frac{2}{3}} - 216 = 0$ involves a power of the variable. Find all real solutions of the equation.

$
\begin{equation}
\begin{aligned}
6x^{\frac{2}{3}} - 216 &= 0 && \text{Add both sides by 216}\\
\\
6x^{\frac{2}{3}} - 216 + 216 &= 0 + 216 && \text{Simplify}\\
\\
6x^{\frac{2}{3}} &= 216 && \text{Divide both sides by 6}\\
\\
\frac{\cancel{6}x^{\frac{2}{3}}}{\cancel{6}} &= \frac{216}{6} && \text{Simplify}\\
\\
x^{\frac{2}{3}} &= 36 && \text{Raise both sides by } \frac{3}{2}\\
\\
\left( x^{\frac{2}{3}} \right)^{\frac{3}{2}} &= \pm (36)^{\frac{3}{2}} && \text{Simplify}\\
\\
x &= \pm (36)^{\frac{3}{2}} && \text{Simplify}\\
\\
x &= \pm \left[ (36)^{\frac{1}{2}} \right]^3 && \text{Simplify}\\
\\
x &= \pm (6)^3 && \text{Simplify}\\
\\
x &= \pm 216
\end{aligned}
\end{equation}
$