Answer
Solution set as $x$-intercepts = $\{ 1,729\}$
The graph of equation $y=x^{\frac{1}{3}}+2x^{\frac{1}{6}}-3$ is shown in Graph$(e)$
Work Step by Step
$y=x^{\frac{1}{3}}+2x^{\frac{1}{6}}-3$
To find $x$ -intercept, $y=0$
$x^{\frac{1}{3}}+2x^{\frac{1}{6}}-3=0$
This equation is equivalent to
$(x^{\frac{1}{6}})^{2}+2x^{\frac{1}{6}}-3=0$
Let $(x^{\frac{1}{6}}) = u$
$u^{2}+2u-3=0$
By factoring,
$(u-1)(u+3)=0$
$u=1$ or $u=-3$
Let $u=1$
Replace $u= (x^{\frac{1}{6}}) $
$ (x^{\frac{1}{6}}) =1$
Raise to the power $6$ on both sides
$ (x^{\frac{1}{6}}) ^6=(1)^{6}$
$x=1$
Let $u=-3$
$ (x^{\frac{1}{6}}) =-3$
$ (x^{\frac{1}{6}}) ^6=(-3)^{6}$
$x=729$
Solution set as $x$-intercepts = $\{ 1,729\}$
The graph of equation $y=x^{\frac{1}{3}}+2x^{\frac{1}{6}}-3$ is shown in Graph$(e)$
