Answer
$y=729$
Work Step by Step
$ y^{\frac{1}{3}}-y^{\frac{1}{6}}-6=0\qquad$...substitute $y^{\frac{1}{6}}$ for $u$ so that $u^{2}=y^{\frac{1}{3}}$
Note that $y^{\frac{1}{6}}$ is the sixth root of $y$, a positive number.
$ u^{2}-u-6=0\qquad$... solve with the Quadratic formula.
$u=\displaystyle \frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$u=\displaystyle \frac{1\pm\sqrt{1+24}}{2}$
$u=\displaystyle \frac{1\pm\sqrt{25}}{2}$
$u=\displaystyle \frac{1\pm 5}{2}$
$u=\displaystyle \frac{1+5}{2}=\frac{6}{2}=3$ or $u=\displaystyle \frac{1-5}{2}=\frac{-4}{2}=-2$
Bring back $y^{\frac{1}{6}}=u$.
$y^{\frac{1}{6}}=3$ or $ y^{\frac{1}{6}}=-2\qquad$...raise both sides of both expressions to the sixth power.
We discard $-2$ because the sixth root of $y$ is nonnegative.
$(y^{\frac{1}{6}})^{6}=3^{6} $
$y=729$
