Ta có : ${{2009}^{2}}={{\left( 2008+1 \right)}^{2}}={{2008}^{2}}+2.2008+1$
Suy ra : $1+{{2008}^{2}}={{2009}^{2}}-2.2008={{2009}^{2}}-2.2009.\frac{2008}{2009}$
Do đó :
$B=\sqrt{1+{{2008}^{2}}+\frac{{{2008}^{2}}}{{{2009}^{2}}}}+\frac{2008}{2009}$

$B=\sqrt{{{2009}^{2}}-2.2009.\frac{2008}{2009}+\frac{{{2008}^{2}}}{{{2009}^{2}}}}+\frac{2008}{2009}$

$B=\sqrt{{{\left( 2009-\frac{2008}{2009} \right)}^{2}}}+\frac{2008}{2009}$

$B=2009-\frac{2008}{2009}+\frac{2008}{2009}$

$B=2009$