I can't seem to solve this one. I tried reversing the signs, which didn't work for me. How do I solve this one?
that won't quite do the job.
$\displaystyle (1 + \sqrt{3} - \sqrt{5})(1 - \sqrt{3} + \sqrt{5}) = 1^2 - (\sqrt{3} - \sqrt{5})^2 $
$\displaystyle = 1 - (3 - 2\sqrt{15} + 5) = -7 + 2\sqrt{15}$, which certainly isn't rational.
i propose instead multiplying top and bottom by:
$\displaystyle 7 + 3\sqrt{3} + \sqrt{5} + 2\sqrt{15}$, since:
$\displaystyle (1 + \sqrt{3} - \sqrt{5})(7 + 3\sqrt{3} + \sqrt{5} + 2\sqrt{15}) = 11$