You didn't distribute when you multiplied the numerator by $\displaystyle\sqrt{{{b}}}\usepackage{cmbright}\Huge\$ . The first two steps are:

$\displaystyle\frac{{{a}+\sqrt{{b}}}}{\sqrt{{{b}}}}\cdot\frac{\sqrt{{{b}}}}{\sqrt{{{b}}}}=\frac{{{a}\cdot\sqrt{{{b}}}+\sqrt{{{b}}}\cdot\sqrt{{{b}}}}}{\sqrt{{{{b}}^{{2}}}}}\usepackage{cmbright}\Huge\$

Multiply the radicals, then simplify each term.