O-Level Surds: worked solution

Question

A cuboid has base area (4 + 2√5) cm² and volume (9 + 5√5) cm³. Without a calculator, find its height in cm in the form ½(a + b√5) with a, b integers.

For a cuboid, volume = base area × height, so height = volume ÷ base area. Height = (9 + 5√5)/(4 + 2√5) cm A surd must not be left in the denominator, so rationalise it: multiply the top and the bottom by the conjugate of the denominator, 4 − 2√5. Height = (9 + 5√5)(4 − 2√5) / [(4 + 2√5)(4 − 2√5)] Numerator: (9 + 5√5)(4 − 2√5) = 36 − 18√5 + 20√5 − 10 × 5 = 36 + 2√5 − 50 = −14 + 2√5 Denominator (difference of two squares): (4 + 2√5)(4 − 2√5) = 4² − (2√5)² = 16 − 4 × 5 = 16 − 20 = −4 Height = (−14 + 2√5)/(−4) Divide top and bottom by −2: = (7 − √5)/2 Height = ½(7 − √5) cm, so a = 7 and b = −1.