Answer
The solution of the equation is $ x=625$.
Work Step by Step
Let us consider the provided equation:
${{x}^{\frac{1}{2}}}-2{{x}^{\frac{1}{4}}}-15=0$.
Assume ${{x}^{\frac{1}{4}}}=t $ and put it into the equation:
$\begin{align}
& {{x}^{\frac{1}{2}}}-2{{x}^{\frac{1}{4}}}-15=0 \\
& {{\left( {{x}^{\frac{1}{4}}} \right)}^{2}}-2{{x}^{\frac{1}{4}}}-15=0 \\
& {{t}^{2}}-2t-15=0 \\
& \left( t-5 \right)\left( t+3 \right)=0
\end{align}$
Further solve the equation as given below:
$\begin{align}
& t=5,-3 \\
& {{x}^{\frac{1}{4}}}=5,-3
\end{align}$
When $ t=5$, the value of $ x $ is
$\begin{align}
& {{x}^{\frac{1}{4}}}=5 \\
& x=625
\end{align}$
When $ t=-3$, the value of $ x $ is
$\begin{align}
& {{x}^{\frac{1}{4}}}=-3 \\
& x=81
\end{align}$
But $ x=81$, does not satisfy the expression ${{x}^{\frac{1}{2}}}-2{{x}^{\frac{1}{4}}}-15=0$.
Thus, the solution of the equation is $ x=625$.
