Evaluate the following if z = 5 – 2i and w = -1 + 3i (i) z + w (ii

1.	Evaluate the following if z = 5 – 2i and w = -1 + 3i (i) z + w (ii) z – iw (iii) 2z + 3w (iv) zw (v) z2 + 2zw+ w2 (vi) (z + w)2
Answer» (i) z = 5 – 2i, w = -1 + 3i z + w = (5 – 2i) + (-1 + 3i) = (5 – 1) + (-2i + 3i) = 4 + i (ii) z – iw = (5 – 2i) – i (-1 + 3i) = 5 – 2i + i + 3 = (5 + 3) + (-2i + i) = 8 – i (iii) 2z + 3w = 2(5 – 2i) + 3 (-1 +3i) = 10 – 4i – 3 + 9i = 7 + 5 i (iv) zw = (5 – 2i) (-1 + 3i) = -5 + 15i + 2i – 6i = -5 + 17i + 6 = 1 + 17i (v) z² + 2zw + w² = (z + w)² [from (i)] = (4 + i)² = 16 – 1 + 8i = 15 + 8i (vi) (z + w)² = 15 + 8z [from (v)]

Evaluate the following if z = 5 – 2i and w = -1 + 3i (i) z + w (ii) z – iw (iii) 2z + 3w (iv) zw (v) z2 + 2zw+ w2 (vi) (z + w)2

Answer»

(i) z = 5 – 2i, w = -1 + 3i

z + w = (5 – 2i) + (-1 + 3i)

= (5 – 1) + (-2i + 3i)

= 4 + i

(ii) z – iw = (5 – 2i) – i (-1 + 3i)

= 5 – 2i + i + 3

= (5 + 3) + (-2i + i)

= 8 – i

(iii) 2z + 3w = 2(5 – 2i) + 3 (-1 +3i)

= 10 – 4i – 3 + 9i

= 7 + 5 i

(iv) zw = (5 – 2i) (-1 + 3i)

= -5 + 15i + 2i – 6i

= -5 + 17i + 6

= 1 + 17i

(v) z² + 2zw + w² = (z + w)² [from (i)]

= (4 + i)²

= 16 – 1 + 8i

= 15 + 8i

(vi) (z + w)²

= 15 + 8z [from (v)]