This question was uploaded on 10/01/22 on social media accounts.
Solution:
\[\color{green} {a+b=1}\]
\[\color{purple} {a^2+b^2=2}\]
\[\color{red} {a^2+b^2=(a+b)^2-2ab}\]
\[\Rightarrow\color{red} {2=1-2ab}\]
\[\Rightarrow\color{red} {ab=-\frac12}\]
\[\color{blue} {a^4+b^4=(a^2+b^2)^2-2(ab)^2}\]
\[\Rightarrow\color{blue} {a^4+b^4=4-\frac12=\frac72}\]
\[\color{fuchsia} {a^8+b^8=(a^4+b^4)^2-2(ab)^4}\]
\[\Rightarrow\color{fuchsia} {a^8+b^8=\frac{49}4-\frac18=\frac{97}8}\]
