Calculate a 43i14i b 1i12i14i c i23i342i2 d pii100pii100Solu

Calculate:

a) (4+3i)/(-1-4i)

b) (1+i)(1-2i)(1-4i)

c) i(2+3i)^3/(4-2i)^2

d) (pi+i)^100/(pi-i)^100

Solution

a. (4+3i)/(-1-4i)

   (4+3i)(-1+4i)/(-1-4i)(-1+4i)

(-4 +16i-3i+12i²)/(1 -4i + 4i -16i²) = (-4 +13i -12)/(1+16) = (-16+13i)/17= -16/17 + 13i/17

b. (1+i)(1-2i)(1-4i)

(1-i -2i²)(1-4i) = (3-i)(1-4i) = 3 -12i -i +4i²= -1 -13i

c. i(2+3i)³/(4-2i)² = i(2+3i)(2+3i)(2+3i)/(4-2i)(4-2i)

                          = i(-46+9i)/(12-16i)

                          = (-46i -9)(12+16i)/(12-16i)(12+16i)

                          = (-552i +736-108-144i)/400

                          = (-696i +628)/400

                          = -696i/400 +628/400 = -87i/50 + 157/100 = 157/100- 87i/50