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.
Best Answer
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.