document.write( "Question 1104069: determine all the indicated roots and represent them graphically :
\n" ); document.write( "a) (4*√2+4*√2i)^1/3 \n" ); document.write( "

Algebra.Com's Answer #718803 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here is a representation of $\"4sqrt%282%29%2Bi4sqrt%282%29\"$
\n" ); document.write( "

$\"4sqrt%282%29%2Bi4sqrt%282%29=r%28cos%28theta%29%2Bi%2Asin%28theta%29%29\"$
\n" ); document.write( "Obviously, $\"tan%28theta%29=4sqrt%282%29%2F4sqrt%282%29=1\"$ , so $\"theta=pi%2F4\"$ .
\n" ); document.write( "In degrees, $\"theta=45%5Eo\"$ .
\n" ); document.write( " $\"r=sqrt%28%284sqrt%282%29%29%5E2%2B%284sqrt%282%29%29%5E2%29=sqrt%2832%2B32%29=sqrt%2864%29=8\"$
\n" ); document.write( "So,
\n" ); document.write( " $\"4sqrt%282%29%2Bi4sqrt%282%29=8%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29\"$ .
\n" ); document.write( "There are $\"3\"$ complex values for $\"%288%28cos%28pi%2F4%29%2Bi%2Asin%28pi%2F4%29%29%29%5E%281%2F3%29\"$ .
\n" ); document.write( "They are complex numbers of the form $\"r%28cos%28theta%29%2Bi%2Asin%28theta%29%29\"$
\n" ); document.write( "such that

.
\n" ); document.write( "So, it will be $\"r%5E3=8\"$ and $\"3theta=pi%2F4%2Bk%282pi%29\"$ for $\"%22k+=+0+%2C+1+%2C+2%22\"$ .
\n" ); document.write( "In degrees, $\"3theta=45%5Eo%2Bk360%5Eo\"$ .
\n" ); document.write( "That means $\"r=2\"$ and
\n" ); document.write( " $\"system%28theta=pi%2F12%2Ctheta=3pi%2F4%2Ctheta=17pi%2F12%29\"$ .
\n" ); document.write( "In degrees,

.
\n" ); document.write( "The answers could be written as
\n" ); document.write( " $\"green%28Z%5B1%5D%29=2cos%2815%5Eo%29%2Bi%2Asin%2815%5Eo%29\"$ , $\"red%28Z%5B2%5D%29=2cos%28135%5Eo%29%2Bi%2Asin%28135%5Eo%29\"$ , and $\"blue%28Z%5B3%5D%29=2cos%28255%5Eo%29%2Bi%2Asin%28255%5Eo%29\"$
\n" ); document.write( "(or you could express the angles in radians).
\n" ); document.write( "They can be represented as
\n" ); document.write( "

\n" ); document.write( " \n" ); document.write( "

\n" );