Question

Докажите,  что $sqrt{2010+2 sqrt{2009} } - sqrt{2010-2 sqrt{2009} }$ целое число и найдите его.

Answer 1

$sqrt{2010+2sqrt{2009}}-sqrt{2010-2sqrt{2009}}=\\=sqrt{2009+2sqrt{2009}+1}-sqrt{2009-2sqrt{2009}+1}=\\=sqrt{(1+sqrt{2009})^2}-sqrt{(1-sqrt{2009})^2}=|1+sqrt{2009}|-|1-sqrt{2009}|=\\=1+sqrt{2009}-(sqrt{2009}-1)=1+1=2$