Đặt \(a=2010^{2009};b=2009^{2009}\)\(\left(a,b>0\right)\)
\(A=\left(a+b\right)^{2010}=\left(a+b\right)^{2009}.\left(a+b\right)\)
\(B=\left(a.2010+b.2009\right)^{2009}=\left[a+2009\left(a+b\right)\right]^{2009}\)
Chia A và B cho \(\left(a+b\right)^{2009}:\)
\(A=a+b;B=\dfrac{\left[a+2009\left(a+b\right)\right]^{2009}}{\left(a+b\right)^{2009}}\)\(=\left(\dfrac{a}{a+b}+2009\right)^{2009}\)
Dễ thấy A<B.
\(B=\left(2010^{2009}.2010+2009^{2009}.2009\right)^{2009}\)
\(B< \left(2010^{2009}.2010+2009^{2009}.2010\right)^{2009}\)
\(B< \left(2010^{2009}+2009^{2009}\right)^{2009}.2010^{2009}\)
\(B< \left(2010^{2009}+2009^{2009}\right)^{2009}.\left(2010^{2009}+2009^{2009}\right)\)
\(B< \left(2010^{2009}+2009^{2009}\right)^{2010}\)
\(\Rightarrow B< A\)
