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Question Number 151205 by EDWIN88 last updated on 19/Aug/21

$$\underbrace{}\overline{)\begin{array}{c}{(2+\sqrt{3})}^x+1={\left(2\sqrt{2+\sqrt{3}}\right)}^x\\ x=?\end{array}}$$

Answered by bramlexs22 last updated on 19/Aug/21

$$\frac{1}{{(2-\sqrt{3})}^{\mathrm{x}}}+1=\frac{2^{\mathrm{x}}}{{(2-\sqrt{3})}^{\frac{\mathrm{x}}{2}}}$$$$\frac{1}{{(2-\sqrt{3})}^{\frac{\mathrm{x}}{2}}}+{(2-\sqrt{3})}^{\frac{\mathrm{x}}{2}}=2^{\mathrm{x}}$$$$\frac{1}{{(2-\sqrt{3})}^{\mathrm{x}}}+{(2-\sqrt{3})}^{\mathrm{x}}=4^{\mathrm{x}}-2$$$${(2+\sqrt{3})}^{\mathrm{x}}+{(2-\sqrt{3})}^{\mathrm{x}}=4^{\mathrm{x}}-2$$$$\mathrm{x}=2,\mathrm{check}\Rightarrow \mathrm{LHS}\equiv {(2+\sqrt{3})}^2+{(2-\sqrt{3})}^2=7+7=14$$$$\mathrm{RHS}\equiv 4^2-2=14$$

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