\(A=2^0+2^1+2^2+2^3+-+2^{2010}\)

\(2A=2^1+2^2+2^3+2^4+-.+2^{2011}\)

\(2A-A=\left(2^1+2^2+2^3+2^4+-.+2^{2011}\right)-\left(2^0+2^1+2^2+2^3+-+2^{2010}\right)\)\(A=2^{2011}-2^0=2^{2011}-1\)

\(A=B=2^{2011}-1\)

\(A=2009.2011=2009.\left(2010+1\right)=2009.2010+2011\)

\(B=2010^2=2010.2010=2010\left(2009+1\right)=2010.2009+2010\)

\(A>B\)

\(A=10^{30}=\left(10^3\right)^{10}=1000^{10}\)

\(B=2^{100}=\left(2^{10}\right)^{10}=1024^{10}\)

\(A< B\)

\(A=333^{444}=\left(333^4\right)^{111}=\left(3^4.111^4\right)^{111}=\left(81.111^4\right)^{111}\)

\(B=444^{333}=\left(444^3\right)^{111}=\left(4^3.111^3\right)^{111}=\left(64.111^3\right)^{111}\)

\(A>B\)

\(A=3^{150}\)

\(B=5^{300}=\left(5^2\right)^{150}=25^{150}\)

\(A< B\)