Question

Choose the correct answer The number $\sqrt{14+6\sqrt{5}}+\sqrt{14-6\sqrt{5}}$
a) is not a rational number b) is a rational number $\ge$ 14
c) simplifies to 5 d) simplifies to 6 (Take positive root only)

Answer 1

$14+6\sqrt{5}=9+5+2\times 3\times \sqrt{5}=3^2+{\sqrt{5}}^2+2\times 3\times \sqrt{5}={(3+\sqrt{5})}^{2}Similarly,14-6\sqrt{5}={(3-\sqrt{5})}^2\sqrt{14+6\sqrt{5}}+\sqrt{14-6\sqrt{5}}=(3+\sqrt{5})+(3-\sqrt{5})=6$