document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=420562>Question 420562</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the multiplicative inverse of<BR>\n" );
document.write( "3+5i<BR>\n" );
document.write( " 8 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #293838 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=293838\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm a little confused by the 8? I'm assuming that it is a typo.<br><BR>\n" );
document.write( "The multiplicative inverse of 3+5i is what you misgh expect it to be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%283%2B5i%29 ALIGN=MIDDLE  ALT=\"1%2F%283%2B5i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "However this is not the proper form. We want something in the standard form:<BR>\n" );
document.write( "a + bi<BR>\n" );
document.write( "where a and b are real numbers.<br><BR>\n" );
document.write( "To put <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%283%2B5i%29%7D%7D+in+standard+form+we+multiply+its+numerator+and+denominator+by+the+complex+conjugate+of+the+denominator.+The+complex+conjugate+of+3%2B5i+is+3-5i.+So+we+multiply+the+numerator+and+denominator+by+3-5i%3A%0D%0A%7B%7B%7B%281%2F%283%2B5i%29%29%28%283-5i%29%2F3-5i%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying the numerator is simple. To multiply the denominator we can use FOIL or the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > pattern. I prefer using the pattern:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-5i%29%2F%28%283%29%5E2-%285i%29%5E2%29 ALIGN=MIDDLE  ALT=\"%283-5i%29%2F%28%283%29%5E2-%285i%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-5i%29%2F%289-25i%5E2%29 ALIGN=MIDDLE  ALT=\"%283-5i%29%2F%289-25i%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-5i%29%2F%289-25%28-1%29%29 ALIGN=MIDDLE  ALT=\"%283-5i%29%2F%289-25%28-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2+=+-1 ALIGN=MIDDLE  ALT=\"i%5E2+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-5i%29%2F%289%2B25%29 ALIGN=MIDDLE  ALT=\"%283-5i%29%2F%289%2B25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-5i%29%2F34 ALIGN=MIDDLE  ALT=\"%283-5i%29%2F34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For standard form we need to split up this fraction:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F34%2B+%28-5i%29%2F34 ALIGN=MIDDLE  ALT=\"3%2F34%2B+%28-5i%29%2F34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F34%2B+%28%28-5%29%2F34%29i ALIGN=MIDDLE  ALT=\"3%2F34%2B+%28%28-5%29%2F34%29i\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );