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Algebra Level 3

Given the expression (1x)(1+x256)(1+x16)(1+x2)(1+x8)(1+x32)(1+x)(1+x4)(1+x128)(1-x)(1+x^{256})(1+x^{16})(1+x^2)(1+x^8)(1+x^{32})(1+x)(1+x^4)(1+x^{128})


The above expression can be written as 1xa1+xb\dfrac{1-x^a}{1+x^b} such that a,bNa,b \in \mathbb{N} , 102a10310^2 \leq a \leq 10^3 and 10b10210 \leq b \leq 10^2.

Find the value of a+b\sqrt{a+b}

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