\(\begin{array}{*{20}{l}}
{{\rm{w}} = {{(1 + {z_1})}^{100}} + {{(1 + {z_2})}^{100}}}\\
{ = {{\left( {{z_1}^2 + 2{z_1} + 1} \right)}^{50}} + {{\left( {{z_2}^2 + 2{z_2} + 1} \right)}^{50}}}\\
\begin{array}{l}
 = {\left( { - 2{z_1} - 4} \right)^{50}} + {\left( { - 2{z_2} - 4} \right)^{50}}{\mkern 1mu} \\
(Do{\mkern 1mu} {z_i}^2 + 4{z_i} + 5 = 0)
\end{array}\\
{ = {2^{50}}{{\left( {{z_1} + 2} \right)}^{50}} + {2^{50}}{{\left( {{z_2} + 2} \right)}^{50}}}\\
{ = {2^{50}}\left[ {{{\left( {{z_1}^2 + 4{z_1} + 4} \right)}^{25}} + {{\left( {{z_2}^2 + 4{z_2} + 4} \right)}^{25}}} \right]}\\
{ = {2^{50}}\left[ {{{\left( { - 1} \right)}^{25}} + {{\left( { - 1} \right)}^{25}}} \right] =  - {2^{51}}.}
\end{array}\)