\[\begin{array}{l}\frac{{\sqrt {{a^3}} - a}}{{a - 2{a^{\frac{1}{2}}} + 1}} = \left[ {a = {{\left( {{a^{\frac{1}{2}}}} \right)}^2}} \right] = \frac{{a\left( {\sqrt a - 1} \right)}}{{{{\left( {{a^{\frac{1}{2}}} - 1} \right)}^2}}} = \\\left[ {\sqrt a = {a^{\frac{1}{2}}}} \right] = \frac{{a\left( {{a^{\frac{1}{2}}} - 1} \right)}}{{{{\left( {{a^{\frac{1}{2}}} - 1} \right)}^2}}} = \frac{a}{{{a^{\frac{1}{2}}} - 1}} = \frac{a}{{\sqrt a - 1}}\end{array}\]

\[\frac{a}{{\sqrt a - 1}}\]