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# Polar Form !!

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What are the solutions to x3 = −4 + 4i in polar form?

6√32cis(19π/12)

3√5cis(23π/12)

6√32cis(2π/3)

6√32cis(π/4)

3√5cis(7π/12)

6√32cis(11π/12)

3√5cis(2π/3)

3√5cis(5π/4)

6√32cis(4π/3)

hey! I'm not asking to do the whole problem. if someone could help me solve/find one, I can do the rest. :)

May 17, 2020

#1
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x =  -4 + 4i

Step 1)  write  -4 + 4i  in  r·cs( theta ) form

r  =  sqrt( (-4)2 + (42) )

theta  =  tan-1( 4 / -4)

that your angle is in the second quadrant>

take the 3rd root of the r-value and divide the angle by 3

Step 3)  To find the other roots:

--  add 2pi/3 to the angle of the first answer (keep r the same)

--  add 2pi/3 to the angle of the second answer (keep r the same)

May 17, 2020
#2
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Why would you take the 3rd root of the r-value? And wouldn't that be 3√32 ? According to the solutions given, wouldn't it be 6, not 3??

And for the angle, I got -45 degrees, which equals -π/4, so if I divided that by 3, then I get -π/12.

Is that right?

auxiarc  May 20, 2020
#3
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r  is  (32)1/2   --->   r1/3  =  [ (32)1/2 ]1/3 =  (32)1/6

When you find theta, you use  tan-1( 4 / -4 )  =  tan-1( -1 )

One of the values of  tan-1( -1 )  is  -45o  or  -pi/4  -- however --

for this problem, since the number is

-4 + 4i -- which is in the second quadrant, you need to use either:

if you are using degrees:  -45o + 180o  =  135o

or, if you are using radians:  - pi/4 + pi  =  3pi/4.

May 20, 2020