Answer
$\left( 81,0 \right)$
Work Step by Step
$f\left( x \right)={{x}^{{}^{1}/{}_{2}}}-{{x}^{{}^{1}/{}_{4}}}-6$.
The x-intercept occurs when$f\left( x \right)=0$,
${{x}^{{}^{1}/{}_{2}}}-{{x}^{{}^{1}/{}_{4}}}-6=0$ …… (1)
Let $u={{x}^{{}^{1}/{}_{4}}}$ and ${{u}^{2}}={{x}^{{}^{1}/{}_{2}}}$,
Substitute the values of $u$ and ${{u}^{2}}$ in equation (1),
${{u}^{2}}-u-6=0$
Now, factor the equation.
$\begin{align}
& {{u}^{2}}-u-6=0 \\
& {{u}^{2}}-3u+2u-6=0 \\
& u\left( u-3 \right)+2\left( u-3 \right)=0 \\
& \left( u-3 \right)\left( u+2 \right)=0
\end{align}$
$\left( u-3 \right)=0$ or $\left( u+2 \right)=0$
If $\left( u-3 \right)=0$
$\begin{align}
& u-3=0 \\
& u=3
\end{align}$
If $\left( u+2 \right)=0$
$\begin{align}
& u+2=0 \\
& u=-2 \\
\end{align}$
Now, replace $u$ with ${{x}^{{}^{1}/{}_{4}}}$
Solve for ${{x}^{{}^{1}/{}_{4}}}=3$,
$\begin{align}
& {{x}^{{}^{1}/{}_{4}}}=3 \\
& {{\left( {{x}^{{}^{1}/{}_{4}}} \right)}^{4}}={{\left( 3 \right)}^{4}} \\
& x=81
\end{align}$
So, the root is $81$. …… (2)
For${{x}^{{}^{1}/{}_{4}}}=-2$,
It has no solution.
Thus, the x-intercept is at $\left( 81,0 \right)$.
