document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208829>Question 1208829</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Note: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When I use the word \"bar\" after the complex number expressions, I simply mean there is a bar over each expression. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Given z = 3 - 4i and w = 8 + 3i, write each expression in the standard form <BR>\n" );
document.write( "a + bi. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1. z • z bar\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2. (z - w) bar\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847305 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52786)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847305\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "Given z = 3 - 4i and w = 8 + 3i, write each expression in the standard form <BR>\n" );
document.write( "a + bi. <BR>\n" );
document.write( "1. z • z bar<BR>\n" );
document.write( "2. (z - w) bar<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "(1)  z • z bar = (3-4i)*(3+4i) = 3*3 - 4*3i + 4*3i - (4i)*(3i) = 9 - 16*i^2 = 9 - 16*(-1) = 9 + 16 = 25.   <U>ANSWER</U>\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     <U>Post solution note</U> :   z* z bar is always real non-negative number (almost always real  <U>positive</U>  number).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(2)  (z - w) = (3-4i) - (8+3i) = (3-8) + (-4i + 3i) = -5 - 7i.\r\n" );
document.write( "\r\n" );
document.write( "     Therefore  (z - w) bar = (-5 - 7i) bar = -5 + 7i.    <U>ANSWER</U>\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "Solved.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );