If $a+b=2$ and $a^{2}+b^{2}=2$, what is the value of $a^{3}+b^{3}$? And $a^{4}+b^{4}$?
According to the identity above, we have $a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)=$ $2(2-a b)$. Therefore, it is sufficient to evaluate $a b$. We have $4=(a+b)^{2}=a^{2}+b^{2}+2 a b=2+2 a b$, hence $a b=1$, and $a^{3}+b^{3}=2$. Next, $4=\left(a^{2}+b^{2}\right)^{2}=a^{4}+b^{4}+2 a^{2} b^{2}=a^{4}+b^{4}+2$, hence $a^{4}+b^{4}=2$.