Calculate $\;$ $\left(\dfrac{x - 1}{x^{\frac{3}{4}} + x^{\frac{1}{2}}}\right) \cdot \left(\dfrac{x^{\frac{1}{2}} + x^{\frac{1}{4}}}{x^{\frac{1}{2}} + 1}\right) \cdot x^{\frac{1}{4}} + 1$ $\;\;\;$ for $\;$ $x = 16$
$\left(\dfrac{x - 1}{x^{\frac{3}{4}} + x^{\frac{1}{2}}}\right) \cdot \left(\dfrac{x^{\frac{1}{2}} + x^{\frac{1}{4}}}{x^{\frac{1}{2}} + 1}\right) \cdot x^{\frac{1}{4}} + 1$
$= \left(\dfrac{x - 1}{x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} + x^{\frac{1}{2}}}\right) \cdot \left(\dfrac{x^{\frac{1}{2}} + x^{\frac{1}{2}} \cdot x^{\frac{-1}{4}}}{x^{\frac{1}{2}} + 1}\right) \cdot x^{\frac{1}{4}} + 1$
$= \left[\dfrac{x - 1}{x^{\frac{1}{2}} \left(x^{\frac{1}{4}} + 1\right)}\right] \cdot \left[\dfrac{x^{\frac{1}{2}} \left(1 + x^{\frac{-1}{4}}\right)}{x^{\frac{1}{2}} + 1}\right] \cdot x^{\frac{1}{4}} + 1$
$= \dfrac{\left(x - 1\right) \cdot \left(1 + \dfrac{1}{x^{\frac{1}{4}}}\right) \cdot x^{\frac{1}{4}}}{\left(x^{\frac{1}{4}} + 1\right) \cdot \left(x^{\frac{1}{2}} + 1\right)} + 1$
$= \dfrac{\left(x - 1\right) \cdot \left(x^{\frac{1}{4}} + 1\right) \cdot x^{\frac{1}{4}}}{x^{\frac{1}{4}} \left(x^{\frac{1}{4}} + 1\right) \left(x^{\frac{1}{2}} + 1\right)} + 1$
$= \dfrac{x - 1}{x^{\frac{1}{2}} + 1} + 1$
$= \dfrac{\left(x - 1\right) \left(x^{\frac{1}{2}} - 1\right)}{\left(x^{\frac{1}{2}} + 1\right) \left(x^{\frac{1}{2}} - 1\right)} + 1$
$= \dfrac{\left(x - 1\right) \left(x^{\frac{1}{2}} - 1\right)}{x - 1} + 1$
$= x^{\frac{1}{2}} - 1 + 1 = x^{\frac{1}{2}}$
$= \left(16\right)^{\frac{1}{2}}$ $\;\;\;$ when $\;$ $x = 16$
$= 4$