## Precalculus (6th Edition) Blitzer

The value of the expression ${{x}^{\frac{1}{2}}}-6{{x}^{\frac{1}{4}}}+8=0$ is $x=16$ or $x=256$.
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 $x=256$.