Question : The square root of $\frac{2+\sqrt{3}}{2}$ is:
Option 1: $\pm \frac{1}{\sqrt{2}}(\sqrt{3}+1)$
Option 2: $\pm \frac{1}{2}(\sqrt{3}-2)$
Option 3: $\text{none}$
Option 4: $\pm \frac{1}{2}(\sqrt{3}+1)$
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Correct Answer: $\pm \frac{1}{2}(\sqrt{3}+1)$
Solution : Given: $\frac{2+\sqrt{3}}{2}$ $= \frac{1}{4}({4+2\sqrt{3}})$ $= \frac{1}{4}[1^2+(\sqrt{3})^2+2×1×\sqrt{3}]$ $= \frac{1}{4}(1+\sqrt{3})^2$ So, the square root of $\frac{2+\sqrt{3}}{2}$ $= \sqrt{\frac{2+\sqrt{3}}{2}}$ $= \sqrt{\frac{1}{4}(1+\sqrt{3})^2}$ $= \pm \frac{1}{2}(1+\sqrt{3})$ Hence, the correct answer is $\pm \frac{1}{2}(\sqrt{3}+1)$ .
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