I missed you, but you don't miss terms !

Notice how you can group the equation to this:

$(1-x)(x + 1)(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1)(x^{32} + 1)(x^{128} + 1)(x^{256} + 1)$

Can you see that the term $(x^{64} + 1)$ has been removed ? So let's multiply and divide by that. $\dfrac{(x - 1)(x + 1)(x^2 + 1)(x^4 + 1)(x^8 + 1)(x^{16} + 1)(x^{32} + 1)(x^{64}+1)(x^{128} + 1)(x^{256} + 1)}{1+x^{64}}$

Now use $(1-x^k)(1+x^k)=1-x^{2k}$ repeatedly to arrive at

$\dfrac {1-x^{512} }{x^{64} + 1}$

This expression satisfies the conditions. Now we need to find the value of

$\sqrt {512 + 64} = 24$

Therefore, 24 is the answer.

Yeah, $x^{64} +1$ is missing so we have $\frac{1-x^{512}}{1+x^{64}}$ . Now, $\sqrt{512+64} = \sqrt{64} \sqrt{8+1} = \sqrt{64}\sqrt{9} = 8 \times 3 = \boxed{24}$

since each (a + b)(a- b ) = a^2 - b^2 now in the given expression for each 1 - x^n there exists 1 + x ^n except 1 + x ^ 64 so divide by 1 + x ^ 64 we get 1 -x^512 / 1+ x^64 hence a = 512 and b = 64 now a + b = 576 sqrt (576) = 24