How to simplify the nested radical $\sqrt{1 - \frac{\sqrt{3}}{2}}$ by hand? I was solving a Mock Mathcounts Contest Mock contest (.pdf) written by a user on the Art of Problem Solving Forums. In problem #24 the only thing I couldn't do by hand was simplify the radical mentioned above. Note that the contest should involve math topics accessible to a high performing Mathcounts middle school student.

How do you simplify $$\sqrt{1 - \frac{\sqrt{3}}{2}}$$


Proposed another way to collection
You can also use formulas
$$\sqrt{a\pm\sqrt b}=\sqrt{\frac{a + \sqrt{a^2-b}}{2}}\pm\sqrt{\frac{a - \sqrt{a^2-b}}{2}}$$
Then
$$\sqrt{1-\frac{\sqrt3}{2}}=\frac12\sqrt{4-\sqrt{12}}=\frac12 \cdot\left(\sqrt{\frac{4 + \sqrt{4^2-12}}{2}}-\sqrt{\frac{4 - \sqrt{4^2-12}}{2}} \right)=$$
$$=\frac12 \left(\sqrt{\frac{4 + \sqrt{4}}{2}}-\sqrt{\frac{4 - \sqrt{4}}{2}} \right)=\frac12 \left(\sqrt{\frac{6}{2}}-\sqrt{\frac{2}{2}} \right)=\frac 12(\sqrt3-1)$$
