1. Let's start by writing the expression clearly: $$\sqrt{2 - \frac{\sqrt{3}}{2}} \times \sqrt{2 + \frac{\sqrt{3}}{2}}$$ 2. We can use the property of square roots that $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$. So, rewrite the expression as: $$\sqrt{\left(2 - \frac{\sqrt{3}}{2}\right) \left(2 + \frac{\sqrt{3}}{2}\right)}$$ 3. Recognize this as a difference of squares: $$ (a - b)(a + b) = a^2 - b^2 $$ where $$a = 2$$ and $$b = \frac{\sqrt{3}}{2}$$. 4. Calculate $$a^2$$: $$2^2 = 4$$ 5. Calculate $$b^2$$: $$\left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}$$ 6. Now, subtract: $$4 - \frac{3}{4} = \frac{16}{4} - \frac{3}{4} = \frac{13}{4}$$ 7. Thus, the expression inside the square root simplifies to $$\frac{13}{4}$$. 8. Now take the square root: $$\sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2}$$ **Final answer:** $$\frac{\sqrt{13}}{2}$$