Question

If z₁ = 3, z₂ = 7i, and z₃ = 5 + 4i, show that (z₁ + z₂)z₃ = z₁z₃ + z₂z₃

Answer 1

z₁ + z₂ = 3 – 7i

(z₁ + z₂)z₃ = (3 – 7i)(5 + 4i)

= 15 + 12i – 35i + 28i²

= 15 + 12i – 35i + 28i

= 43 – 23i  .......(1)

z₁z₃ = 3(5 + 4i) = 15 + 12i

z₂z₃ = – 7i(5 + 4i) = – 35i – 28i²

= – 35i + 28

z₁z₃ + z₂z₃ = (15 + 12i) + (– 35i + 28)

= 15 + 12i – 35i + 28

= 43 – 23i  .......(2)

∴ From 1 and 2

(z₁ + z₂)z₃ = z₁z₃ + z₂z₃

Hence proved.