1. **State the problem:** Find the value of the expression $$(3 + \sqrt{3\sqrt{3}})(3 - \sqrt{\sqrt{27}}).$$ 2. **Simplify the radicals:** - Simplify $\sqrt{3\sqrt{3}}$: $$\sqrt{3\sqrt{3}} = \sqrt{3 \times 3^{1/2}} = \sqrt{3^{1 + 1/2}} = \sqrt{3^{3/2}} = 3^{3/4}.$$ - Simplify $\sqrt{\sqrt{27}}$: $$\sqrt{\sqrt{27}} = \sqrt{27^{1/2}} = 27^{1/4} = (3^3)^{1/4} = 3^{3/4}.$$ 3. **Rewrite the expression using the simplified radicals:** $$(3 + 3^{3/4})(3 - 3^{3/4}).$$ 4. **Use the difference of squares formula:** $$(a + b)(a - b) = a^2 - b^2,$$ where $a = 3$ and $b = 3^{3/4}$. 5. **Calculate each square:** $$a^2 = 3^2 = 9,$$ $$b^2 = (3^{3/4})^2 = 3^{3/2} = 3^{1.5} = 3 \times \sqrt{3} = 3\sqrt{3}.$$ 6. **Substitute back:** $$9 - 3\sqrt{3}.$$ 7. **Final answer:** $$9 - 3\sqrt{3}.$$ This corresponds to option A.