Choose the correct answer. The number \(\sqrt{14+6\sqrt5} + \sqrt{14-6

1.	Choose the correct answer. The number \(\sqrt{14+6\sqrt5} + \sqrt{14-6\sqrt5}\)(a) is not a rational number (b) is a rational number ≥ 14 (c) simplifies to 5 (d) simplifies to 6. (Take positive root only)
Answer» (d) simplifies to 6. (Take positive root only) \(\sqrt{14+6\sqrt5} = \sqrt{9+5+2\times3\times\sqrt5}\) = \(\sqrt{(3)^2+(\sqrt5)^2+2\times3\sqrt5}\) = \(\sqrt{(3+\sqrt5)^2}\) = 3 + \(\sqrt5\) Similarly, \(\sqrt{14+6\sqrt5} = 3 - \sqrt5 \) ∴ \(\sqrt{14+6\sqrt5} + \sqrt{14-6\sqrt5} \) = 3 + \(\sqrt5\) + 3 - \(\sqrt5\) = 6. ∴ (d) is the correct answer.

Choose the correct answer. The number \(\sqrt{14+6\sqrt5} + \sqrt{14-6\sqrt5}\)(a) is not a rational number (b) is a rational number ≥ 14 (c) simplifies to 5 (d) simplifies to 6. (Take positive root only)

Answer»

(d) simplifies to 6. (Take positive root only)

\(\sqrt{14+6\sqrt5} = \sqrt{9+5+2\times3\times\sqrt5}\)

= \(\sqrt{(3)^2+(\sqrt5)^2+2\times3\sqrt5}\)

= \(\sqrt{(3+\sqrt5)^2}\) = 3 + \(\sqrt5\)

Similarly, \(\sqrt{14+6\sqrt5} = 3 - \sqrt5 \)

∴ \(\sqrt{14+6\sqrt5} + \sqrt{14-6\sqrt5} \) = 3 + \(\sqrt5\) + 3 - \(\sqrt5\) = 6.

∴ (d) is the correct answer.