Question 1: Simplify the expression \( \frac{3}{\sqrt{5}} \) by rationalising the denominator.

Answer: \frac{3\sqrt{5}}{5}

Question 2: Rationalise the denominator of \( \frac{7}{\sqrt{3}} \).

Answer: \frac{7\sqrt{3}}{3}

Question 3: Express \( \frac{5}{\sqrt{2}} \) with a rational denominator.

Answer: \frac{5\sqrt{2}}{2}

Question 4: Simplify \( \frac{9}{\sqrt{10}} \) after rationalising the denominator.

Answer: \frac{9\sqrt{10}}{10}

Question 5: Calculate and simplify \( \frac{8}{\sqrt{7}} \).

Answer: \frac{8\sqrt{7}}{7}

Question 6: Rationalise and simplify \( \frac{12}{\sqrt{6}} \).

Answer: \frac{12\sqrt{6}}{6} = 2\sqrt{6}

Question 7: Construct a triangle on the grid with one side length \( 3\sqrt{2} \) and another \( 5\sqrt{2} \), forming a right angle at the vertex. (Draw on grid)

Answer: Student completes drawing.

Question 8: Solve for \( x \): \( \frac{x}{\sqrt{3}} = 4 \).

Answer: x = 4\sqrt{3}

Question 9: A student mistakenly multiplies numerator and denominator of \( \frac{6}{\sqrt{2}} \) by \( \sqrt{2} \) instead of rationalising properly. Identify and correct the mistake.

Answer: Incorrect: \( \frac{6\sqrt{2}}{\sqrt{2}\times\sqrt{2}} = \frac{6\sqrt{2}}{2} \). Correct: Rationalise properly to get \( \frac{6\sqrt{2}}{2} = 3\sqrt{2} \).

Question 10: Evaluate and simplify \( \frac{15}{\sqrt{3}} \).

Answer: 5\sqrt{3}

Question 11: Challenge: Rationalise \( \frac{10}{\sqrt{5} + 1} \) using conjugates.

Answer: \frac{10(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)} = \frac{10\sqrt{5} - 10}{5 - 1} = \frac{10\sqrt{5} - 10}{4} = \frac{5\sqrt{5} - 5}{2}