By nested radical, I mean an expression of the form $\sqrt{a+b\sqrt{n}}$ where a, b and n are positive integers and n is not a perfect square.
I wrote a computer program that randomly generated pairs of nested radicals (with a common value of n) such that neither nested radical could denest but their product could. One example is:
$$\sqrt{8+2\sqrt{11}} \times \sqrt{13+2\sqrt{11}} = 7+3\sqrt{11}$$
But when I randomly generated pairs of nested radicals (with a common value of n) such that neither nested radical could denest but hoping their sum could, I couldn’t find any. Do any such pairs exist? If so, I’d like a few examples and some guidance about how to find more. If such pairs don’t exist, I’d like to know why not.
It is not possible. Suppose the following sum denested:
$$ \sqrt{a+b\sqrt{n}}+\sqrt{c+d\sqrt{n}}= u + v \sqrt{n} \tag{1} $$
Then the difference of the two radicals on the LHS would also denest (assuming $\,u,v\,$ rational and $\,\sqrt{n}\,$ irrational, so $\,u^2 - v^2n \ne 0\,$):
$$ \begin{align} \sqrt{a+b\sqrt{n}} - \sqrt{c+d\sqrt{n}} &= \frac{\left(a+b\sqrt{n}\right)-\left(c+d\sqrt{n}\right)}{\sqrt{a+b\sqrt{n}}+\sqrt{c+d\sqrt{n}}} \\ &= \frac{(a-c) + (b-d)\sqrt{n}}{u+v\sqrt{n}} \color{blue}{\cdot \frac{u - v\sqrt{n}}{u-v\sqrt{n}}} \\ &= \frac{(a-c)u-(b-d)vn + \left((b-d)u-(a-c)v\right)\sqrt{n}}{u^2 - v^2n} \\ &= u' + v' \sqrt{n} \tag{2} \end{align} $$
Adding $\,(1)+(2)\,$ gives $\,\sqrt{a+b\sqrt{n}} = \dfrac{1}{2}\big((u+u') + (v+v')\sqrt{n}\big)\,$, and subtracting $\,(1)-(2)\,$ gives $\,\sqrt{c+d\sqrt{n}} = \dfrac{1}{2}\big((u-u') + (v-v')\sqrt{n}\big)\,$, so both radicals denest.