Question 1: Simplify the expression \( \frac{3}{\sqrt{2}} \) by rationalising the denominator.

Answer: 3\sqrt{2}/2

Question 2: Express \( \frac{5}{\sqrt{5}} \) in simplified form with a rationalised denominator.

Answer: 5\sqrt{5}/5 = \sqrt{5}

Question 3: Calculate and rationalise \( \frac{7}{\sqrt{3}} \).

Answer: 7\sqrt{3}/3

Question 4: Simplify \( \frac{8}{\sqrt{8}} \).

Answer: 8\sqrt{8}/8 = \sqrt{8} = 2\sqrt{2}

Question 5: Construct a mathematical expression with a denominator involving \( \sqrt{b} \) that simplifies to a whole number after rationalising.

Answer: Solution: \( \frac{4}{\sqrt{4}} \) simplifies to 2.

Question 6: Rationalise and simplify \( \frac{9}{\sqrt{6}} \).

Answer: 9\sqrt{6}/6

Question 7: In a real-world context, a farmer needs to build a square plot of land with a diagonal length of \( \sqrt{50} \) meters. Calculate the length of one side of the square after rationalising the necessary expressions.

Answer: Side length = \( \sqrt{50}/\sqrt{2} = \sqrt{50} \times \sqrt{2}/2 = \sqrt{100}/2 = 10/2 = 5 \) meters.

Question 8: Explain why rationalising the denominator is useful in simplifying expressions like \( \frac{6}{\sqrt{3}} \).

Answer: Rationalising removes the radical from the denominator, making the expression easier to interpret, compare, and perform further calculations.

Question 9: Given \( \frac{a}{\sqrt{b}} \), with \( a=12 \) and \( b=18 \), rationalise the denominator and simplify your answer.

Answer: 12\sqrt{18}/18 = 12\times \sqrt{18}/18 = (12/18)\sqrt{18} = (2/3)\times 3\sqrt{2} = 2\sqrt{2}

Question 10: Identify and correct the mistake in the following: \( \frac{5}{\sqrt{7}} = \frac{5\sqrt{7}}{7} \).

Answer: Incorrect. The correct rationalisation is \( \frac{5}{\sqrt{7}} = \frac{5\sqrt{7}}{7} \). The mistake is in the initial step; the correction is to multiply numerator and denominator by \( \sqrt{7} \).

Question 11: Challenge: Rationalise \( \frac{7}{\sqrt{2} + 1} \) and simplify the result.

Answer: Multiply numerator and denominator by \( \sqrt{2} - 1 \). Numerator: \( 7(\sqrt{2} - 1) = 7\sqrt{2} - 7 \) Denominator: \( (\sqrt{2} + 1)(\sqrt{2} - 1) = 2 - 1 = 1 \) Result: \( 7\sqrt{2} - 7 \).

Question 12: Evaluate \( \frac{10}{\sqrt{5}} \) and express in simplest form.

Answer: 10\sqrt{5}/5 = 2\sqrt{5}