Perform the indicated operations and simplify the expression.

$$\frac{1}{2}x^{-1/4}=\frac{1}{2x^{0.25}}=\frac{1}{2\sqrt[4]{x}}$$

Perform the indicated operations and simplify the expression.

$$\small{\text{
$
\left( x^{\frac{1}{4}} + 1 \right)
\cdot
\textcolor[rgb]{1,0,0}{\left( \dfrac{1}{4}x^{-\frac{1}{4}} \right)}
-
\left( x^{\frac{1}{4}} - 1 \right)
\cdot
\textcolor[rgb]{1,0,0}{\left( \dfrac{1}{4}x^{-\frac{1}{4}} \right)}
$}}\\
\small{\text{
$
=
\textcolor[rgb]{1,0,0}{\left( \dfrac{1}{4}x^{-\frac{1}{4}} \right)}
\cdot
\left[
\left( x^{\frac{1}{4}} + 1 \right)
-
\left( x^{\frac{1}{4}} - 1 \right)
\right]
$}}\\\\
\small{\text{
$
=
\left( \dfrac{1}{4}x^{-\frac{1}{4}} \right)
\cdot
\left(
\not{x^{\frac{1}{4}}} + 1
- \not{x^{\frac{1}{4}}} + 1
\right)
$}}\\\\
\small{\text{
$
=\left( \dfrac{2}{4}x^{-\frac{1}{4}} \right)
$}}\\\\
\small{\text{
$
=\left( \dfrac{1}{2}x^{-\frac{1}{4}} \right)
$}}$$