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Simplify $$\sqrt{1 + \frac{\sqrt{3}}{2}} + \sqrt{1 - \frac{\sqrt{3}}{2}}$$

May 6, 2020

#1
+24951
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Simplify

$$\sqrt{1 + \dfrac{\sqrt{3}}{2}} + \sqrt{1 - \dfrac{\sqrt{3}}{2}}$$

$$\begin{array}{|rcll|} \hline && \mathbf{ \sqrt{1 + \dfrac{\sqrt{3}}{2}} + \sqrt{1 - \dfrac{\sqrt{3}}{2}} } \\\\ &=& \sqrt{ \left(\sqrt{1 + \dfrac{\sqrt{3}}{2}} + \sqrt{1 - \dfrac{\sqrt{3}}{2}}\right)^2 } \\\\ &=& \sqrt{ \left(\sqrt{1 + \dfrac{\sqrt{3}}{2}}\right)^2 +2 \sqrt{1 + \dfrac{\sqrt{3}}{2}}\sqrt{1 - \dfrac{\sqrt{3}}{2}} + \left(\sqrt{1 - \dfrac{\sqrt{3}}{2}}\right)^2 } \\\\ &=& \sqrt{ 1 + \dfrac{\sqrt{3}}{2} + 2\sqrt{ \left(1 + \dfrac{\sqrt{3}}{2}\right) \left(1 - \dfrac{\sqrt{3}}{2}\right) } + 1 - \dfrac{\sqrt{3}}{2} } \\\\ &=& \sqrt{ 2 + 2\sqrt{ \left(1 + \dfrac{\sqrt{3}}{2}\right) \left(1 - \dfrac{\sqrt{3}}{2}\right) } } \\\\ &=& \sqrt{ 2 + 2\sqrt{ 1^2-\left( \dfrac{\sqrt{3}}{2} \right)^2 } } \\\\ &=& \sqrt{ 2 + 2\sqrt{ 1 - \dfrac{3}{4} } } \\\\ &=& \sqrt{ 2 + 2\sqrt{ \dfrac{1}{4} } } \\\\ &=& \sqrt{ 2 + 2\cdot \dfrac{1}{2} } \\\\ &=& \sqrt{ 2 + 1 } \\\\ &=& \mathbf{\sqrt{3}} \\ \hline \end{array}$$

May 6, 2020