How exactly does \((a^4 + 4)\) help?

Algebra Level 4

\[\large{\frac{\bigg(2^4 + \frac14\bigg)\bigg(4^4 + \frac14\bigg)\ldots\bigg(198^4 + \frac14\bigg)\bigg(200^4 + \frac14\bigg)}{\bigg(1^4 + \frac14\bigg)\bigg(3^4 + \frac14\bigg)\ldots\bigg(197^4 + \frac14\bigg)\bigg(199^4 + \frac14\bigg)} = ?} \]

HINT: Write the expression \((a^4 + 4)\) as a product of two factors, where each one of those factors is equal to a sum of two squares.

Bonus: Simpify the following product.\[\large{\displaystyle \prod_{k=1}^n \frac{(2k)^4+\frac14}{(2k-1)^4+\frac14}}\]

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