24. \( 2 \sqrt{2 a}(3 \sqrt{12 a}-\sqrt{27 a}) \) 25. \( 3 \sqrt[3]{3 a}\left(4 \sqrt[3]{9 a^{2}}+5 \sqrt[3]{7 a^{2}}\right) \) \( 26 .(\sqrt{x}+2 \sqrt{y})(2 \sqrt{x}-\sqrt{y}) \) \( 27(2 \sqrt{x}-3 \sqrt{y})(3 \sqrt{x}+5 \sqrt{y}) \)

Let's explore how each of these mathematical expressions can lead to real-world applications and some handy problem-solving tips! First Line: The first expression, \( 2 \sqrt{2 a}(3 \sqrt{12 a}-\sqrt{27 a}) \), resembles formulas used in physics, particularly in calculating areas or volumes involving square roots, like the area of a circular segment when considering gravitational forces. The simplification could illustrate how to manage complex physical scenarios in engineering fields, ensuring quick and accurate calculations! Second Line: When tackling these types of expressions, a common mistake is forgetting to simplify each square root term thoroughly. For example, breaking down \( \sqrt{27 a} \) into \( 3 \sqrt{3a} \) before continuing can reveal essential insights. Always remember to look for common factors or perfect squares to simplify your work!