Answer
$x=729$ and $x=27$
Work Step by Step
$x^{1/2}-3x^{1/3}=3x^{1/6}-9$
Let $x^{1/6}$ be equal to $u$
$u=x^{1/6}$
$u^{2}=x^{1/3}$
$u^{3}=x^{1/2}$
Rewrite the original equation using $u$:
$u^{3}-3u^{2}=3u-9$
Take all terms to the left side:
$u^{3}-3u^{2}-3u+9=0$
Factor this equation by grouping terms:
$(u^{3}-3u^{2})-(3u-9)=0$
Take out common factor $u^{2}$ from the first parentheses and common factor $3$ from the second one:
$u^{2}(u-3)-3(u-3)=0$
Rewrite by factoring out common factor $u-3$:
$(u-3)(u^{2}-3)=0$
Set both factors equal to $0$ and solve each individual equation for $u$:
$u-3=0$
$u=3$
$u^{2}-3=0$
$u^{2}=3$
$u=\pm\sqrt{3}$
Substitute $u$ back to $x^{1/6}$ and solve for $x$:
$u=3$
$x^{1/6}=3$
$x=3^{6}$
$x=729$
$u=\sqrt{3}$
$x^{1/6}=\sqrt{3}$
$x=(\sqrt{3})^{6}$
$x=27$
$u=-\sqrt{3}$
$x^{1/6}=-\sqrt{3}$
$x=(-\sqrt{3})^{6}$
$x=27$
The solutions found are $729$ and $27$. The original equation is true for both of them. The final answer is:
$x=729$ and $x=27$
