1. **State the problem:** Simplify the expressions $$(2\sqrt{3})^2$$, $$(2 - \sqrt{3})^2$$, and $$(2 + \sqrt{3})(2 - \sqrt{3})$$ and explain how they differ. 2. **Formula and technique:** - For expressions like $$(a)^2$$, use the rule $$(a)^2 = a \times a$$. - For binomials squared, use the formula $$(a - b)^2 = a^2 - 2ab + b^2$$. - For the product of conjugates, use $$(a + b)(a - b) = a^2 - b^2$$. 3. **Simplify $$(2\sqrt{3})^2$$:** $$ (2\sqrt{3})^2 = (2\sqrt{3}) \times (2\sqrt{3}) $$ $$ = 2 \times 2 \times \sqrt{3} \times \sqrt{3} $$ $$ = 4 \times 3 = 12 $$ 4. **Simplify $$(2 - \sqrt{3})^2$$ using binomial square formula:** $$ (2 - \sqrt{3})^2 = 2^2 - 2 \times 2 \times \sqrt{3} + (\sqrt{3})^2 $$ $$ = 4 - 4\sqrt{3} + 3 $$ $$ = 7 - 4\sqrt{3} $$ 5. **Simplify $$(2 + \sqrt{3})(2 - \sqrt{3})$$ using product of conjugates:** $$ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 $$ $$ = 4 - 3 = 1 $$ 6. **Explanation:** - The first expression is a square of a single term involving a square root. - The second is a square of a binomial, requiring expansion using the binomial square formula. - The third is a product of conjugates, which simplifies to the difference of squares. **Final answers:** $$ (2\sqrt{3})^2 = 12 $$ $$ (2 - \sqrt{3})^2 = 7 - 4\sqrt{3} $$ $$ (2 + \sqrt{3})(2 - \sqrt{3}) = 1 $$