(This question might have incorrect tags)
I was playing with different expressions of factorization. Here's what I was doing. We all know that
$$\begin{align*} a^2-b^2&=(a+b)(a-b)\\[5pt] a-b&=(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})\\[5pt] &=(\sqrt{a}+\sqrt{b})(\sqrt[4]{a}+\sqrt[4]{b})(\sqrt[4]{a}-\sqrt[4]{b})\\[5pt] \end{align*}$$
Now if we continue this process infinitely then we get the formula:
$$a-b=\prod_{k=1}^{\infty}(a^{2^{-k}}+b^{2^{-k}})$$
and
$$\prod_{k=1}^{n}(a^{2^{-k}}+b^{2^{-k}})=\frac{a-b}{a^{2^{-n}}-b^{2^{-n}}}$$
But if I input this thing on wolfram alpha I get nothing.
And also if I input the second expression then I also get nothing.
So, are these formulas correct?
The mistake in your expression of the product results from missing the difference factor.
$a-b=(a^{2^{-n}}-b^{2^{-n}})\prod_{k=1}^n {a^{2^{-k}}+b^{2^{-k}}}$