2-3-x-1-2-3-x-2-x-x-

Question Number 161868 by cortano last updated on 23/Dec/21

{(\sqrt{2 + \sqrt{3}})}^{x} + \frac{1}{{(\sqrt{2 + \sqrt{3}})}^{x}} = 2^{x}

x = ?

Answered by Rasheed.Sindhi last updated on 23/Dec/21

a = \sqrt{2 + \sqrt{3}}, \frac{1}{a} = \frac{1}{\sqrt{2 + \sqrt{3}}} \times \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{2 - \sqrt{3}}}

= \frac{\sqrt{2 - \sqrt{3}}}{\sqrt{4 - 3}} = \sqrt{2 - \sqrt{3}}

a^{2} + \frac{1}{a^{2}} = {(\sqrt{2 + \sqrt{3}})}^{2} + {(\sqrt{2 - \sqrt{3}})}^{2}

= 2 + \sqrt{3} + 2 - \sqrt{3} = 4 = 2^{2}

a^{2} + \frac{1}{a^{2}} = 2^{2}

x = 2