\[\prod\limits_{n = 1}^k {\left( {{x^{{2^n}}} + {y^{{2^n}}}} \right)} = \left\{ \begin{array}{l}\frac{{{x^{{2^{k + 1}}}} - {y^{{2^{k + 1}}}}}}{{{x^2} - {y^2}}},{\text{ if }}x \ne y\\\frac{{{2^k} \cdot {x^{{2^{k + 1}}}}}}{{{x^2}}},{\text{ if }}x = y\end{array} \right.\]