Question 1: Simplify and rationalise the denominator of the expression \( \frac{5}{\sqrt{3}} \).

Answer: 5\sqrt{3}/3

Question 2: Express \( \frac{7}{\sqrt{2}} \) in the form \( \frac{a}{\sqrt{b}} \) and rationalise the denominator.

Answer: 7\sqrt{2}/2

Question 3: A square has a side length of \( \frac{4}{\sqrt{5}} \) units. Rationalise the expression for its side length and calculate the approximate decimal value.

Answer: Approx. 4\sqrt{5}/5 ≈ 1.79

Question 4: Simplify the rational expression \( \frac{8}{\sqrt{6}} \) into the form \( \frac{a}{\sqrt{b}} \) and rationalise the denominator.

Answer: 8\sqrt{6}/6

Question 5: Solve for \( x \) if \( \frac{x}{\sqrt{7}} = \frac{3\sqrt{7}}{7} \).

Answer: 3

Question 6: Express \( \frac{9}{\sqrt{8}} \) in the form \( \frac{a}{\sqrt{b}} \) and rationalise.

Answer: 9\sqrt{8}/8

Question 7: The length of a rectangular garden is \( \frac{6}{\sqrt{2}} \) meters, and the width is \( \frac{3}{\sqrt{2}} \) meters. Rationalise both expressions and find the area of the garden.

Answer: Area = (6\sqrt{2}/2) \times (3\sqrt{2}/2) = 6 \times 3/2 = 9

Question 8: Construct a triangle on the grid with sides of length \( \frac{5}{\sqrt{2}} \), \( \frac{7}{\sqrt{2}} \), and \( \frac{8}{\sqrt{2}} \). Rationalise the lengths before drawing.

Answer: All sides become 5\sqrt{2}/2, 7\sqrt{2}/2, 8\sqrt{2}/2 respectively.

Question 9: Identify the common mistake made when rationalising \( \frac{10}{\sqrt{5}} \), and correct it.

Answer: Incorrectly multiplied numerator and denominator by \( \sqrt{5} \) without simplifying. Correct by multiplying numerator and denominator by \( \sqrt{5} \), resulting in \( 10\sqrt{5}/5 = 2\sqrt{5} \).