Question

Please answer with proper steps

Answer 1

Answer:

$n=\frac{31}{32}$  Option D) is the correct answer.

Step-by-step explanation:

Given that, $\sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3} } } } } = 3^{n}$  .  .  .  .  .  .  .  .  . (i)

$=>3\sqrt{3\sqrt{3\sqrt{3\sqrt{3} } } } = 3^{2n}$  [squaring both sides]

$=>\sqrt{3\sqrt{3\sqrt{3\sqrt{3} } } } = 3^{2n-1}$  [∵ $\frac{a^m}{a^n} = a^{m-n}$]

$=>3\sqrt{3\sqrt{3\sqrt{3} } } = 3^{2(2n-1)}$  [squaring both sides]

$=>\sqrt{3\sqrt{3\sqrt{3} } } = 3^{2(2n-1)-1} = 3^{4n-3}$

$=>3\sqrt{3\sqrt{3} } = 3^{2(4n-3)}$  [squaring both sides]

$=>\sqrt{3\sqrt{3} } = 3^{2(4n-3)-1} = 3^{8n-7}$

$=>3\sqrt{3} = 3^{2(8n-7)}$   [squaring both sides]

$=>\sqrt{3} = 3^{2(8n-7)-1} = 3^{16n-15}$

$=>3 = 3^{2(16n-15)}$  [squaring both sides]

∴ 32n - 30 = 1

=> 32n = 31

∴ $n=\frac{31}{32}$  Option D) is the correct answer.