Simplify each of the following and express it in the form (a + ib) :

(–3 + 5i)³

Given: (-3 + 5i)³

We know that,

(-a + b)³ = - a³ + 3a²b – 3ab² + b³ …(i)

So, on replacing a by 3 and b by 5i in eq. (i), we get

-(3)³ + 3(3)²(5i) – 3(3)(5i)² + (5i)³

= -27 + 3(9)(5i) – 3(3)(25i²) + 125i³

= -27 + 135i – 225i² + 125i³

= -27 + 135i – 225 × (-1) + 125i × i²

= -27 + 135i + 225 – 125i [∵ i² = -1]

= 198 + 10i