Let $${S_k} = {\alpha ^k} + {\beta ^k} + {\gamma ^k},$$    then \[\Delta = \left| {\begin{array}{*{20}{c}} {{S_0}}&{{S_1}}&{{S_2}}\\ {{S_1}}&{{S_2}}&{{S_3}}\\ {{S_2}}&{{S_3}}&{{S_4}} \end{array}} \right|\]    is equal to

Solution :
\[\begin{array}{l} \Delta = \left| {\begin{array}{*{20}{c}} {{S_0}}&{{S_1}}&{{S_2}}\\ {{S_1}}&{{S_2}}&{{S_3}}\\ {{S_2}}&{{S_3}}&{{S_4}} \end{array}} \right|\\ = \,\left| {\begin{array}{*{20}{c}} {1 + 1 + 1}&{\alpha + \beta + \gamma }&{{\alpha ^2} + {\beta ^2} + {\gamma ^2}}\\ {\alpha + \beta + \gamma }&{{\alpha ^2} + {\beta ^2} + {\gamma ^2}}&{{\alpha ^3} + {\beta ^3} + {\gamma ^3}}\\ {{\alpha ^2} + {\beta ^2} + {\gamma ^2}}&{{\alpha ^3} + {\beta ^3} + {\gamma ^3}}&{{\alpha ^4} + {\beta ^4} + {\gamma ^4}} \end{array}} \right| \end{array}\]
The above determinant can be expressed as product of two determinants. Thus,
\[\Delta = \left| {\begin{array}{*{20}{c}} 1&1&1\\ \alpha &\beta &\gamma \\ {{\alpha ^2}}&{{\beta ^2}}&{{\gamma ^2}} \end{array}} \right|\,\left| {\begin{array}{*{20}{c}} 1&1&1\\ \alpha &\beta &\gamma \\ {{\alpha ^2}}&{{\beta ^2}}&{{\gamma ^2}} \end{array}} \right| = {\left[ {\left( {\beta - \alpha } \right)\left( {\gamma - \alpha } \right)\left( {\gamma - \beta } \right)} \right]^2}\]