document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=131945>Question 131945</A>: <I><SPAN STYLE=\"word-break: break-all;\"> could someone please help me with this problem? <BR>\n" );
document.write( "(3-i) / (1+i)   I know you use the conjugate and multiply by (3-i) / (1+I) I think,but I don't think I did something right.Thanks for your help in advance.<BR>\n" );
document.write( "Sincerely,LH </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #96342 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=96342\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A conjugate of a complex number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi ALIGN=MIDDLE  ALT=\"a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is formed by changing the sign on the imaginary part.  So the conjugate of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bbi ALIGN=MIDDLE  ALT=\"a%2Bbi\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-bi ALIGN=MIDDLE  ALT=\"a-bi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "To simplify an expression with a complex denominator, you need to multiply the entire expression by 1 in the form of the conjugate of the denominator divided by itself.  That means if you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bbi%29%2F%28c%2Bdi%29 ALIGN=MIDDLE  ALT=\"%28a%2Bbi%29%2F%28c%2Bdi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you want to multiply by 1 in the form of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c-di%29%2F%28c-di%29 ALIGN=MIDDLE  ALT=\"%28c-di%29%2F%28c-di%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Your problem:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-i%29+%2F+%281%2Bi%29 ALIGN=MIDDLE  ALT=\"%283-i%29+%2F+%281%2Bi%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The conjugate of the denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-i%29 ALIGN=MIDDLE  ALT=\"%281-i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so you need to multiply the entire expression by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-i%29%2F%281-i%29 ALIGN=MIDDLE  ALT=\"%281-i%29%2F%281-i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, thus:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283-i%29%2F%281%2Bi%29%29%28%281-i%29%2F%281-i%29%29 ALIGN=MIDDLE  ALT=\"%28%283-i%29%2F%281%2Bi%29%29%28%281-i%29%2F%281-i%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now multiply numerator times numerator and denominator times denominator just like multiplying any other pair of fractions.\r<BR>\n" );
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document.write( "Using FOIL, the  numerator product becomes: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B3i-i%2B1=4%2B2i ALIGN=MIDDLE  ALT=\"3%2B3i-i%2B1=4%2B2i\"   DISABLED_style= \"max-width:90%; height:auto;\" > (remember <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;\" > so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-i%2Ai=-%28i%5E2%29=-%28-1%29=1 ALIGN=MIDDLE  ALT=\"-i%2Ai=-%28i%5E2%29=-%28-1%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "The denominator product is easier because, by using the conjugate, you have the factors of the difference of two squares, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bi%29%281-i%29=1-i%5E2=1-%28-1%29=2 ALIGN=MIDDLE  ALT=\"%281%2Bi%29%281-i%29=1-i%5E2=1-%28-1%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Putting the pieces back together we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2B2i%29%2F2=2%2Bi ALIGN=MIDDLE  ALT=\"%284%2B2i%29%2F2=2%2Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Done.\r<BR>\n" );
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