Question 1 1 Simplify the following surds: \( 1.1 \quad \sqrt{75} \) \( 1.2 \quad \sqrt{50}+\sqrt{18} \) \( 1.3 \quad \frac{\sqrt{48}}{\sqrt{3}} \) \( 1.4 \quad \sqrt{27}+\sqrt{12} \) Question 2 2 Perform the following operations with surds \( 2.1 \quad \sqrt{18}+\sqrt{32} \) \( 2.2 \quad \sqrt{75}-\sqrt{12} \) \( 2.3 \quad \sqrt{6} \times \sqrt{3} \) \( 2.4 \quad \sqrt{8} \times \sqrt{2} \) \( 2.5 \quad \sqrt{18}+\sqrt{8} \) \( 2.6 \quad \sqrt{20}+\sqrt{45}-\sqrt{80} \) Question 3 3 Rationalize the denominator of the following expressions: \( 3.1 \quad \frac{5}{\sqrt{3}} \)

To simplify \( \sqrt{75} \), we notice that \( 75 = 25 \times 3 \). Thus, \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \). Similarly, \( \sqrt{50}+\sqrt{18} \) simplifies to \( 5\sqrt{2} + 3\sqrt{2} = 8\sqrt{2} \) after breaking them down into \( 25 \times 2 \) and \( 9 \times 2 \) respectively. Rationalizing \( \frac{5}{\sqrt{3}} \) involves multiplying both the numerator and denominator by \( \sqrt{3} \). This gives \( \frac{5\sqrt{3}}{3} \). This process makes the denominator a rational number, which is often preferred in mathematics for clarity and simplicity in further calculations. In the magical world of math, surds are like hidden gems waiting to be uncovered! Simplifying them allows you to find their true beauty, revealing simpler roots or clean expressions. Embrace the adventure of numbers, where each simplification is a treasure hunt leading to clarity in calculations. To dive deeper, consider exploring how surds are used in geometry, particularly in finding lengths of sides in right triangles using the Pythagorean theorem. Alternatively, biographies of mathematicians like Euclid or modern figures who tackled irrational numbers provide exciting contexts to these tricky elements of math!