Find all complex numbers z such that z^4=-4
Note: All solutions should be expressed in the form a+bi, where a and b are real numbers.Please write your answer in complete sentences.
Thanks in Advance!
z^2 = 2i
z = +- sqrt 2i = 1+i or 1 -i or -1 -i or -1 + i
EP is off-line; I'll try to explain it in a way that makes sense to me.
To find the four fourth roots of -4: z = (-4)1/4
I'm assuming that you are familiar with r·cis( theta ) form; if not, this may not make much sense.
If you graph z = -4 on the complex plane, you'll notic that its distance from the origin is 4;
this means that r = 4
You'll also notice that it's directly west; therefore, its angle is 180o (if you wish, you can change this into radians).
So, the complex number -4 = -4 + 0i = 4cis(180o)
To find the primary fourth root, find the fourth root of r and divide the angle by 4.
So, the primary fourth root of -4 is 41/4cis( 180o/4 ) = sqrt(2)·cis( 45o )
To find the other fourth roots, divide 360o by 4 to get 90o and add this amount onto the angle
of the primary fourt root three times:
So, your four fourth roots are:
sqrt(2)·cis( 45o )
sqrt(2)·cis( 45o + 90o ) = sqrt(2)·cis( 135o )
sqrt(2)·cis( 135o + 90o ) = sqrt(2)·cis( 225o )
sqrt(2)·cis( 225o + 90o ) = sqrt(2)·cis( 315o )
sqrt(2)·cis( 45o ) = sqrt(2)·cos( 45o ) + i·sqrt(2)·sin( 45o ) = 1 + 1i
sqrt(2)·cis( 135o ) = -1 + i
sqrt(2)·cis( 225o ) = -1 - i
sqrt(2)·cis( 315o ) = 1 - i