Question 1178763
<br>
{{{16i = 0+16i = 16cis(pi/2)}}}<br>
Use deMoivre's Theorem on the last form to find the 4th roots<br>
(1) Find the "primary" root.  To find the nth root of a number in a*cis(theta) form, take the nth root of a, and divide the angle theta by n.<br>
{{{(16cis(pi/2))^(1/4) = (16^(1/4))*cis(pi/8) = 2*cis(pi/8)}}}<br>
(2) Find the other roots.  The n n-th roots of a number all have the same magnitude, and they are distributed around the complex plane in intervals of (2pi)/n.<br>
{{{2pi/4 = pi/2}}}<br>
The 4th roots are at intervals of pi/2 in the complex plane.  Starting with the "primary" root of 2*cis(pi/8), the four 4th roots of 16i are<br>
2cis(pi/8)
2cis(5pi/8)
2cis(9pi/8)
2cis(13pi/8)<br>