Significant Squareroots

$\begin{aligned} \frac 1{1+\sqrt 3+\sqrt 5} & = \frac \blue{1+\sqrt 3 - \sqrt 5}{(1+\sqrt 3+\sqrt 5) \blue{(1+\sqrt 3-\sqrt 5)}} \\ & = \frac {1+\sqrt 3 - \sqrt 5}{2\sqrt 3-1} \\ & = \frac {(1+\sqrt 3 - \sqrt 5)\blue{(2\sqrt 3+1)}}{(2\sqrt 3-1)\blue{(2\sqrt 3+1)}} \\ & = \frac {7+3\sqrt 3 - \sqrt 5 - 2\sqrt{15}}{11} \end{aligned}$

Therefore $P = \dfrac 7{11}$ and $A+B = 7 + 11 = \boxed{18}$ .

$\frac1{1+\sqrt{3}+\sqrt{5}}=\frac{(1+\sqrt{3}-\sqrt{5})(1-\sqrt{3}+\sqrt{5})(-1+\sqrt{3}+\sqrt{5})}{(1+\sqrt{3}+\sqrt{5})(1+\sqrt{3}-\sqrt{5})(1-\sqrt{3}+\sqrt{5})(-1+\sqrt{3}+\sqrt{5})}=\frac{7+3\sqrt{3}-\sqrt{5}-2\sqrt{15}}{11}$

(Note that $P,Q,R,S$ are rational numbers, or there will be infinite solutions, so perhaps you should add this in your problem)