Maths Olympiad Prep

Library / /208 of 520

Algebra Difficulty 2.8 Junior Find the answer

With a rational denominator, the expression 22+3−5\frac {\sqrt {2}}{\sqrt {2} + \sqrt {3} - \sqrt {5}} is equivalent to:
(A) 3+6+156\textbf{(A)}\ \frac {3 + \sqrt {6} + \sqrt {15}}{6}(B) 6−2+106\textbf{(B)}\ \frac {\sqrt {6} - 2 + \sqrt {10}}{6}(C) 2+6+1010\textbf{(C)}\ \frac{2+\sqrt{6}+\sqrt{10}}{10}(D) 2+6−106\\ \textbf{(D)}\ \frac {2 + \sqrt {6} - \sqrt {10}}{6}(E) none of these\textbf{(E)}\ \text{none of these}

Multiple choice: answer with the letter of the option you want.

Solution

Let k=2+3k=\sqrt{2}+\sqrt{3}
Then 2k−5  ⟹  2(k+5)k2−52  ⟹  2k+10k2−5  ⟹  2(2+3)+10(2+3)2−5  ⟹  2+6+1026  ⟹  26+6+6012  ⟹  6+3+156A\frac{\sqrt{2}}{k-\sqrt{5}}\implies \frac{\sqrt{2}(k+\sqrt{5})}{k^2-\sqrt{5}^2}\implies\frac{\sqrt{2}k+\sqrt{10}}{k^2-5}\implies \frac{\sqrt{2}(\sqrt{2}+\sqrt{3})+\sqrt{10}}{(\sqrt{2}+\sqrt{3})^2-5}\implies \frac{2+\sqrt{6}+\sqrt{10}}{2\sqrt{6}}\implies\frac{2\sqrt{6}+6+\sqrt{60}}{12}\implies \frac{\sqrt{6}+3+\sqrt{15}}{6}\fbox{A}

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.