Let z=sqrt3-i and w=-2-2i Part I: convert z and w to polar form Part II: calcula

Question

Let z=sqrt3-i and w=-2-2i Part I: convert z and w to polar form Part II: calculate wz using De Moivre's theorem. Express answer in polar (r(costheta +isintheta)) form. Part III: calculate z^2. Express answer in rectangular (a+bi) form. Part IV: calculate w^4. Express answer in polar (r(costheta + isintheta)) form. Let z=sqrt3-i and w=-2-2i Part I: convert z and w to polar form Part II: calculate wz using De Moivre's theorem. Express answer in polar (r(costheta +isintheta)) form. Part III: calculate z^2. Express answer in rectangular (a+bi) form. Part IV: calculate w^4. Express answer in polar (r(costheta + isintheta)) form. Part I: convert z and w to polar form Part II: calculate wz using De Moivre's theorem. Express answer in polar (r(costheta +isintheta)) form. Part III: calculate z^2. Express answer in rectangular (a+bi) form. Part IV: calculate w^4. Express answer in polar (r(costheta + isintheta)) form.

Explanation / Answer

z=sqrt3-i and w=-2-2i

i) Polar form .Find modulsu and argument of each.

z = re^ix ; r = |z| = sqrt(3 +1) = 2

x = tan^-1(-1/sqrt3) = 2pi- pi/6 = 11pi/6

z = 2e^i(11pi/6)

w = re^ix ; |w| = sqrt(2^2 +2^2) = 2sqrt2

x = tan^-1(-2/-2) = 5pi/4

w = 2sqrt(2)*e^i(5pi/4)

ii) wz = 2e^i(11pi/6) *2sqrt(2)*e^i(5pi/4)

= 4sqrt(2)e^(11pi/6 +5pi/4)

= 4sqrt(2)e^i(37pi/8)

= 4sqrt(cos(37pi/8) + i*sin(37pi/8) )

iii) z^2 = [ 2e^i(11pi/6)]^2

= 4e^i(11pi/3)

= 4(cos11pi/3 +isin11pi/3)

= 2 - i*13.46

iv) w^4 = [2sqrt(2)*e^i(5pi/4)]^4

= 64(cos5pi/4 + i*sin5pi/4)

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