1. **State the problem:** Calculate the following complex number expressions given: $z_1 = -2 + 4i$, $z_2 = 3 + 6i$, $z_3 = -7 + 4i$, $z_4 = 3 + 2i$, $z_5 = -1 - 2i$, $z_6 = -4 - 2i$ 2. **Recall multiplication of complex numbers:** $ (a + bi)(c + di) = (ac - bd) + (ad + bc)i $ 3. **Calculate each expression step-by-step:** (1) $z_1 z_2 + z_2 z_3 - z_5$ $z_1 z_2 = (-2 + 4i)(3 + 6i) = (-2)(3) - (4)(6) + [(-2)(6) + (4)(3)]i = -6 - 24 + (-12 + 12)i = -30 + 0i = -30$ $z_2 z_3 = (3 + 6i)(-7 + 4i) = 3(-7) - 6(4) + [3(4) + 6(-7)]i = -21 - 24 + (12 - 42)i = -45 - 30i$ Sum: $-30 + (-45 - 30i) - (-1 - 2i) = (-30 - 45 + 1) + (0 - 30 + 2)i = -74 - 28i$ (2) $z_1 z_3 - z_4 + z_5$ $z_1 z_3 = (-2 + 4i)(-7 + 4i) = (-2)(-7) - 4(4) + [(-2)(4) + 4(-7)]i = 14 - 16 + (-8 - 28)i = -2 - 36i$ Sum: $(-2 - 36i) - (3 + 2i) + (-1 - 2i) = (-2 - 3 - 1) + (-36 - 2 - 2)i = -6 - 40i$ (3) $z_1 - 3 z_2 + 4 z_3 z_6$ Calculate $3 z_2 = 3(3 + 6i) = 9 + 18i$ Calculate $z_3 z_6 = (-7 + 4i)(-4 - 2i) = (-7)(-4) - 4(-2) + [(-7)(-2) + 4(-4)]i = 28 + 8 + (14 - 16)i = 36 - 2i$ Calculate $4 z_3 z_6 = 4(36 - 2i) = 144 - 8i$ Sum: $(-2 + 4i) - (9 + 18i) + (144 - 8i) = (-2 - 9 + 144) + (4 - 18 - 8)i = 133 - 22i$ (4) $z_1 + 3 z_2 z_5 - z_6$ Calculate $z_2 z_5 = (3 + 6i)(-1 - 2i) = 3(-1) - 6(2) + [3(-2) + 6(-1)]i = -3 - 12 + (-6 - 6)i = -15 - 12i$ Calculate $3 z_2 z_5 = 3(-15 - 12i) = -45 - 36i$ Sum: $(-2 + 4i) + (-45 - 36i) - (-4 - 2i) = (-2 - 45 + 4) + (4 - 36 + 2)i = -43 - 30i$ (5) $2 z_3 - 5 z_4 z_5 - 7 z_6$ Calculate $2 z_3 = 2(-7 + 4i) = -14 + 8i$ Calculate $z_4 z_5 = (3 + 2i)(-1 - 2i) = 3(-1) - 2(2) + [3(-2) + 2(-1)]i = -3 - 4 + (-6 - 2)i = -7 - 8i$ Calculate $-5 z_4 z_5 = -5(-7 - 8i) = 35 + 40i$ Calculate $-7 z_6 = -7(-4 - 2i) = 28 + 14i$ Sum: $(-14 + 8i) + (35 + 40i) + (28 + 14i) = (-14 + 35 + 28) + (8 + 40 + 14)i = 49 + 62i$ (6) $3 z_1 - 2 z_7 z_3 - 6 z_4 z_5 - 7 z_6$ Note: $z_7$ is not given, so cannot compute expressions involving $z_7$. Skipping (6), (7). (8) $z_1 z_2 - z_3 - z_4 + 5 z_6$ Recall $z_1 z_2 = -30$ from (1) Calculate $- z_3 = -(-7 + 4i) = 7 - 4i$ Calculate $- z_4 = -(3 + 2i) = -3 - 2i$ Calculate $5 z_6 = 5(-4 - 2i) = -20 - 10i$ Sum: $-30 + (7 - 4i) + (-3 - 2i) + (-20 - 10i) = (-30 + 7 - 3 - 20) + (-4 - 2 - 10)i = -46 - 16i$ (9) $z_1 z_2 z_3 - z_2 z_3 z_5$ Calculate $z_1 z_2 z_3 = (z_1 z_2) z_3 = (-30)(-7 + 4i) = 210 - 120i$ Calculate $z_2 z_3 z_5 = (z_2 z_3) z_5 = (-45 - 30i)(-1 - 2i)$ $= (-45)(-1) - (-30)(-2) + [(-45)(-2) + (-30)(-1)]i = 45 - 60 + (90 + 30)i = -15 + 120i$ Difference: $(210 - 120i) - (-15 + 120i) = 210 + 15 - 120i - 120i = 225 - 240i$ (10) $z_1 + 5 z_6 - 3 z_4 z_6$ Calculate $5 z_6 = -20 - 10i$ Calculate $z_4 z_6 = (3 + 2i)(-4 - 2i) = 3(-4) - 2(2) + [3(-2) + 2(-4)]i = -12 - 4 + (-6 - 8)i = -16 - 14i$ Calculate $-3 z_4 z_6 = -3(-16 - 14i) = 48 + 42i$ Sum: $(-2 + 4i) + (-20 - 10i) + (48 + 42i) = (-2 - 20 + 48) + (4 - 10 + 42)i = 26 + 36i$ (11) $3 z_5 - 2 z_2 z_4 - z_3$ Calculate $3 z_5 = 3(-1 - 2i) = -3 - 6i$ Calculate $z_2 z_4 = (3 + 6i)(3 + 2i) = 3(3) - 6(2) + [3(2) + 6(3)]i = 9 - 12 + (6 + 18)i = -3 + 24i$ Calculate $-2 z_2 z_4 = -2(-3 + 24i) = 6 - 48i$ Calculate $- z_3 = 7 - 4i$ Sum: $(-3 - 6i) + (6 - 48i) + (7 - 4i) = (-3 + 6 + 7) + (-6 - 48 - 4)i = 10 - 58i$ (12) $4 z_1 - 5 z_2 z_6 - 8 z_6$ Calculate $4 z_1 = 4(-2 + 4i) = -8 + 16i$ Calculate $z_2 z_6 = (3 + 6i)(-4 - 2i) = 3(-4) - 6(2) + [3(-2) + 6(-4)]i = -12 - 12 + (-6 - 24)i = -24 - 30i$ Calculate $-5 z_2 z_6 = -5(-24 - 30i) = 120 + 150i$ Calculate $-8 z_6 = -8(-4 - 2i) = 32 + 16i$ Sum: $(-8 + 16i) + (120 + 150i) + (32 + 16i) = (-8 + 120 + 32) + (16 + 150 + 16)i = 144 + 182i$ (13) $3 z_1 z_2 z_3 + z_6$ Recall $z_1 z_2 z_3 = 210 - 120i$ Calculate $3 z_1 z_2 z_3 = 3(210 - 120i) = 630 - 360i$ Sum: $(630 - 360i) + (-4 - 2i) = 626 - 362i$ (14) $4 z_1 z_2 + 3 z_2 z_3 + 7 z_1$ Recall $z_1 z_2 = -30$, $z_2 z_3 = -45 - 30i$ Calculate $4 z_1 z_2 = 4(-30) = -120$ Calculate $3 z_2 z_3 = 3(-45 - 30i) = -135 - 90i$ Calculate $7 z_1 = 7(-2 + 4i) = -14 + 28i$ Sum: $-120 + (-135 - 90i) + (-14 + 28i) = (-120 - 135 - 14) + (-90 + 28)i = -269 - 62i$ (15) $z_2 + z_3 + 2 z_2 z_6$ Recall $z_2 = 3 + 6i$, $z_3 = -7 + 4i$, $z_2 z_6 = -24 - 30i$ Calculate $2 z_2 z_6 = 2(-24 - 30i) = -48 - 60i$ Sum: $(3 + 6i) + (-7 + 4i) + (-48 - 60i) = (3 - 7 - 48) + (6 + 4 - 60)i = -52 - 50i