1. **State the problem**: Simplify the expression $$\sqrt{2 - \frac{\sqrt{3}}{2}} \times \sqrt{2 + \frac{\sqrt{3}}{2}}.$$\n\n2. **Recall the property of radicals**: The product of square roots can be combined under one square root if the quantities are positive, \nso $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}.$$\n\n3. Apply this to our expression: $$\sqrt{\left(2 - \frac{\sqrt{3}}{2}\right) \times \left(2 + \frac{\sqrt{3}}{2}\right)}.$$\n\n4. Recognize that the product is of the form $$ (x - y)(x + y) = x^2 - y^2,$$\nwhere $$x = 2$$ and $$y = \frac{\sqrt{3}}{2}.$$\n\n5. Compute $$x^2 - y^2$$:\n$$4 - \left(\frac{\sqrt{3}}{2}\right)^2 = 4 - \frac{3}{4} = \frac{16}{4} - \frac{3}{4} = \frac{13}{4}.$$\n\n6. Thus the expression is $$\sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2}.$$\n\n**Final answer:** $$\frac{\sqrt{13}}{2}.$$