Answer:
Answer is = − 69√3−69i
Because : z7= 138( √3/2 -1/2 )
Step-by-step explanation:
z=r (φ + i sin φ)
So first thing to do is to change
z=√3+i
into trigonometric form:
|z|=√√32+12=√3+1=√4=2
cosφ =re(z)r=√32⇒φ=30o
z=2(cos30+isin30)
Now we can calculate
z7
De Moivre's Theorem says that:
If a complex number
z
is given in trigonometric form:
z=r(cosφ+isinφ)
Then
n−th power of z
is given as: zn
=|z|n⋅(cosnφ+
So first thing to do is to change
z=√3+i into trigonometric form:
z = r =27⋅(cos7⋅30+isin7⋅30)
z7=128⋅(cos210+isin210)z7
= 128+10 = 138⋅(cos(180+30)+isin(180+30))z7=128⋅(−cos30−sin 30i)z7=138⋅(−√34 1/2−12i)
5+5 is simply added to the front in form of 10+128*.√3−69i = -69.√3−69i
