Complex Product Computation
Let $ a = e^{4 \pi i/5} $ and $ b = e^{2 \pi i/17} $. Compute the value of the product
\[(a + b)(a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]
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- $\frac{a}{b}$
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Solution
We see the factors $ a + b, $ $ a^2 + b^2, $ and $ a^4 + b^4 $. Knowing that $ a^5 = 1 $ and $ b^{17} = 1, $ we can write
\begin{align*}
a^3 + b^8 &= a^8 + b^8, \\
a + b^{16} &= a^{16} + b^{16}, \\
a^2 + b^{15} &= a^{32} + b^{32}, \\
a^4 + b^{13} &= a^{64} + b^{64}, \\
a^3 + b^9 &= a^{128} + b^{128}.\end{align*}Hence, the given product is equal to
\begin{align*}
&(a + b)(a^2 + b^2)(a^4 + b^4)(a^8 + b^8)(a^{16} + b^{16})(a^{32} + b^{32})(a^{64} + b^{64})(a^{128} + b^{128}) \\
&= \frac{a^2 - b^2}{a - b} \cdot \frac{a^4 - b^4}{a^2 - b^2} \dotsm \frac{a^{256} - b^{256}}{a^{128} - b^{128}} \\
&= \frac{a^{256} - b^{256}}{a - b} = \frac{a - b}{a - b} = \boxed{1}.\end{align*}
Freeflows