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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

where is the figure

stethere all not eveanp-by-step explanation: