Understand the concept of a surd as an irrational number represented using a square root or a radical sign.

A rectangular garden bed has length \( (3 + \sqrt{5}) \) metres and width \( (2 + \sqrt{5}) \) metres. Determine the exact area of the garden bed in the form \( a + b\sqrt{5} \), where \( a \) and \( b \) are integers. Evaluate the reasonableness of the claim that this area is greater than 18 square metres by substituting your exact area into an appropriate inequality.

Simplify √98 - √32 + √50, expressing your answer in the form a√b where a and b are integers and b is as small as possible.

Simplify √72 + √32 and express your answer in the form a√b, where a and b are positive integers and b has no perfect square factors other than 1.

A rectangle has a length of \( 3 + 2\sqrt{5} \) metres and a width of \( 4 - \sqrt{5} \) metres. (a) Determine the perimeter of the rectangle, expressing your answer in the form \( a + b\sqrt{5} \), where \( a \) and \( b \) are integers. (2 marks) (b) Determine the area of the rectangle, expressing your answer in the form \( c + d\sqrt{5} \), where \( c \) and \( d \) are integers. (2 marks) (c) Rationalise the denominator of \( \frac{1}{2 + 5\sqrt{5}} \) and express your answer in the form \( p + q\sqrt{5} \), where \( p \) and \( q \) are rational numbers. (1 mark)