Evaluate (5+5sqrt(3)i)^7 using DeMoivre’s theorem.

Write your answer in rectangular form. Show all your work.

jscare May 22, 2020

#1**0 **

Evaluate: ( 5 + 5·sqrt(3)·i )^{7}

1) Write 5 + 5·sqrt(3)·i in r·cis( theta ) form:

If you have a + bi use r = sqrt(a^{2} + b^{2}) and theta = tan^{-1}( b/a ):

r = sqrt( ( 5 )^{2} + ( 5·sqrt(3) )^{2} = sqrt( 25 + 75 = sqrt( 100 ) = 10

theta = tan^{-1}( 5·sqrt(3) / 5 ) = 60^{o} [If you are to use radians, use pi/3.]

[You need to check which quadrant you should use; since both 5 and 5·sqrt(3) are

positive, the angle is in the first quadrant; so 60^{o} is the correct angle.]

2) You now have: 5 + 5·sqrt(3)·i = 10·cis( 60^{o} )

To raise to a power, raise the constant to that power and multiply the angle by the power.

[ 10·cis( 60^{o} ) ]^{7} = 10^{7} · cis( 60^{o} · 7 ) = 10^{7} · cis( 420^{o} ) = 10^{7} · cis( 60^{o} )

3) To write this in rectangular form, use: x = r·cos( theta ) and y = r·sin( theta )

x = 10^{7}·cos( 60^{o} ) = 10^{7}· (1/2) = 5·10^{6}

y = 10^{7}·sin 60^{o} ) = 10^{7}· (sqrt(3)/2) = 5·sqrt(3)·10^{6}

---> 5·10^{6} + 5·sqrt(3)·10^{6}·i

geno3141 May 22, 2020