Solution: The original formula = (√ 2-1)+(√ 3-√ 2)+(√ 4-√ 3)+...+(√ 2008-√ 2007)+(√ 2009-√ 2008) =-65438+.

Not simplification, but the denominator is rationality, and the method is;

1/[√(a+ 1)+√a]=[√(a+ 1)-√a]/{[√(a+ 1)+√a]]} =√(a+ 1)-√a

For example:1/(√ 2009+√ 2008) = (√ 2009-√ 2008)/[(√ 2009+√ 2008)]

=(√2009-√2008)/(2009-2008)=√2009-√2008.

The denominator is the formula: (a+b)(a-b)=a? -B? .