College Algebra, Chapter 1, 1.5, Section 1.5, Problem 46

Find all real solutions of the equation $\displaystyle \sqrt{x} - 3 \sqrt[4]{x} - 4 = 0$


$
\begin{equation}
\begin{aligned}

\sqrt{x} - 3 \sqrt[4]{x} - 4 =& 0
&& \text{Given}
\\
\\
(\sqrt[4]{x})^2 - 3 \sqrt[4]{x - 4} =& 0
&& \text{Let } w = \sqrt[4]{x}
\\
\\
(w - 4)(w + 1)=& 0
&& \text{Factor}
\\
\\
w - 4 =& 0 \text{ and } w + 1 = 0
&& \text{Zero Product Property}
\\
\\
w =& 4 \text{ and } w = -1
&& \text{Solve for } w
\\
\\
\sqrt[4]{x} =& 4 \text{ and } \sqrt[4]{x} = -1
&& \text{Substitute } w = \sqrt[4]{x}
\\
\\
(\sqrt[4]{x})^4 =& (4)^4 \text{ and } (\sqrt[4]{x})^4 = (-1)^4
&& \text{Raise both sides by } 4
\\
\\
x =& 256 \text{ and } x = 1
&& \text{Solve for } x
\\
\\
x =& 256
&& \text{The only solution for the equation } \sqrt{x} - 3 \sqrt[4]{x} - 4 = 0

\end{aligned}
\end{equation}
$