Chapter 1, Equations and Graphs - Section 1.6 - Solving Other Types of Equations - 1.6 Exercises - Page 139: 65

$x=27$ and $x=729$

Let $y=x^{1/6}$. The equation would be $y^3-3y^2=3y-9$. Solve for $y$: $y^3-3y^2-3y+9=0$ (Factorize) $(y-3)(y^2-3)=0$ $y-3=0$ or $y^2-3=0$ $y=3$ or $y^2=3$ $y=3$ or $y=\pm\sqrt{3}$ Solve for $x$: For $y=3$, $x^{1/6}=3\to x=3^6=729$ For $y=-\sqrt{3}$, $x^{1/6}=-\sqrt{3}$ (Impossible) For $y=\sqrt{3}$, $x^{1/6}=\sqrt{3}\to x=\sqrt{3}^6=27$