2-
Simplify the following:
3+i (2 i-4) (6-7 i)
Factor 2 out of 2 i-4 giving 2 (i-2):
i×2 (i-2) (6-7 i)+3
(i-2) (6-7 i) = -2×6-2 (-7 i)+i×6+i (-7 i) = -12+14 i+6 i+7 = -5+20 i:
i×2 20 i-5+3
Factor 5 out of 20 i-5 giving 5 (4 i-1):
i×2×5 (4 i-1)+3
2×5 = 10:
10 i (4 i-1)+3
i (4 i-1) = -4-i:
10 -i-4+3
Factor -1 from -i-4:
10×-(4+i)+3
-10 (i+4) = -40-10 i:
-10 i-40+3
-40-10 i+3 = -10 i+(3-40) = -37-10 i:
-(10 i)-37
Factor -1 from -(10 i)-37:
Answer: | -(37+10 i)
3-
Simplify the following:
4-2 i/(-10)+5 i×3-i/(-3)+i
Multiply numerator and denominator of i/(-10) by -1:
4-2×(-i)/10+5 i×3-i/(-3)+i
(-1)^2 = 1:
4+2 i/10+5 i×3-i/(-3)+i
2/10 = 2/(2×5) = 1/5:
4+i/5+5 i×3-i/(-3)+i
5×3 = 15:
4+i/5+15 i-i/(-3)+i
Multiply numerator and denominator of -i/(-3) by -1:
4+i/5+15 i+i/3+i
Put each term in 4+i/5+15 i+i/3+i over the common denominator 15: 4+i/5+15 i+i/3+i = 60/15+(3 i)/15+(225 i)/15+(5 i)/15+(15 i)/15:
60/15+(3 i)/15+(225 i)/15+(5 i)/15+(15 i)/15
60/15+(3 i)/15+(225 i)/15+(5 i)/15+(15 i)/15 = (60+3 i+225 i+5 i+15 i)/15:
(60+3 i+225 i+5 i+15 i)/15
Add like terms. 60+3 i+225 i+5 i+15 i = 60+248 i:
(60+248 i)/15
Factor 4 out of 60+248 i giving 4 (15+62 i):
Answer: | 4 (15+62 i)/15
4-
Simplify the following:
3 i+1/2-(1-3 i)/(3)+2
Put each term in 3 i+1/2-(1-3 i)/3+2 over the common denominator 6: 3 i+1/2-(1-3 i)/3+2 = (18 i)/6+3/6-(2-6 i)/6+12/6:
(18 i)/6+3/6+(6 i-2)/6+12/6
Factor 2 out of 6 i-2 giving 2 (3 i-1):
(18 i)/6+3/6+2 (3 i-1)/6+12/6
(18 i)/6+3/6+(2 (3 i-1))/6+12/6 = (18 i+3+(6 i-2)+12)/6:
(18 i+3-2+6 i+12)/6
18 i+3-2+6 i+12 = (3-2+12)+(18 i+6 i) = 13+24 i:
Answer: | 13+24 i/6
