Answer
The value of the expression ${{x}^{\frac{1}{2}}}-6{{x}^{\frac{1}{4}}}+8=0$ is $ x=16$ or $ x=256$.
Work Step by Step
Rewrite the provided equation as given below:
$\begin{align}
& {{x}^{\frac{1}{2}}}-6{{x}^{\frac{1}{4}}}+8=0 \\
& {{x}^{\frac{1}{2}}}+8=6{{x}^{\frac{1}{4}}}
\end{align}$
Then, squaring both sides, and get:
$\begin{align}
& x+16{{x}^{\frac{1}{2}}}+64=36{{x}^{\frac{1}{2}}} \\
& x+64=20{{x}^{\frac{1}{2}}}
\end{align}$
Again, square both sides:
$\begin{align}
& {{x}^{2}}+128x+{{64}^{2}}=400x \\
& {{x}^{2}}-272x+4096=0 \\
& {{x}^{2}}-256x-16x+4096=0 \\
& x\left( x-256 \right)-16\left( x-256 \right)=0
\end{align}$
Therefore, one has
$\left( x-256 \right)\left( x-16 \right)=0$
Thus, the value of x is $ x=16$ or $ x=256$.
