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# DeMoivre’s theorem.

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

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

May 22, 2020

### 2+0 Answers

#1
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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(a2 + b2)  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 )  =  60o   [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 60o is the correct angle.]

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

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

[ 10·cis( 60o ) ]7  =  107 · cis( 60o · 7 )  =  107 · cis( 420o )  =  107 · cis( 60o )

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

x  =  107·cos( 60o )  =  107· (1/2)  =  5·106

y  =  107·sin 60o )  =  107· (sqrt(3)/2)  =  5·sqrt(3)·106

--->     5·106  +  5·sqrt(3)·106·i

May 22, 2020
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thank you so muchh!!!!!

jscare  May 22, 2020