document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=421979>Question 421979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me simplify this equation: (4-2i)/(7+3i) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #294610 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=294610\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your expression is (4-2i) / (7 + 3i)\r<BR>\n" );
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document.write( "I initially thought this was a multiplication, but I now see it's a division.\r<BR>\n" );
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document.write( "you would want to remove the i terms from the denominator.\r<BR>\n" );
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document.write( "if you multiply both numerator and denominator by (7-3i), that should do it.\r<BR>\n" );
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document.write( "your expression becomes:\r<BR>\n" );
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document.write( "(4-2i) * (7-3i) divided by (7+3i) * (7-3i)\r<BR>\n" );
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document.write( "(7+3i) * (7-3i) would be equal to 49 - 21i + 21i - 9i^2\r<BR>\n" );
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document.write( "the middle terms cancel out and you are left with 49 -9i^2\r<BR>\n" );
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document.write( "the i's are treated like any other variable until the end, at which time they are finally processed based on the rules for i processing to be described after all is said and done.\r<BR>\n" );
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document.write( "anyway, your denominator is 49 - 9i^2\r<BR>\n" );
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document.write( "you would multiply your numerator out to get:\r<BR>\n" );
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document.write( "(4-2i) * (7-3i) = 28 - 12i - 14i + 6i^2\r<BR>\n" );
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document.write( "combine like terms to get your numerator equal to 28 - 26i + 6i^2\r<BR>\n" );
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document.write( "your expression now has become:\r<BR>\n" );
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document.write( "28 - 26i + 6i^2 divided by 49 - 9i^2\r<BR>\n" );
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document.write( "now you process the i's based on the rules.\r<BR>\n" );
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document.write( "i = i<BR>\n" );
document.write( "i^2 = -1<BR>\n" );
document.write( "i^3 = -i<BR>\n" );
document.write( "i^4 = 1\r<BR>\n" );
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document.write( "this is a cyclical pattern that repeats every 4 exponents.\r<BR>\n" );
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document.write( "i^5 = i<BR>\n" );
document.write( "i^6 = -1<BR>\n" );
document.write( "i^7 = -i<BR>\n" );
document.write( "i^8 = 1\r<BR>\n" );
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document.write( "i^9 = i<BR>\n" );
document.write( "i^10 = -1<BR>\n" );
document.write( "i^11 = -i<BR>\n" );
document.write( "i^12 = 1\r<BR>\n" );
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document.write( "etc.\r<BR>\n" );
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document.write( "your expression is, once again:\r<BR>\n" );
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document.write( "28 - 26i + 6i^2 divided by 49 - 9i^2\r<BR>\n" );
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document.write( "since i^2 is equal to -1, you change your expression to become:\r<BR>\n" );
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document.write( "28 - 26i + 6*(-1) divided by 49 - 9*(-1)\r<BR>\n" );
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document.write( "this simplifies to:\r<BR>\n" );
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document.write( "28 - 26i - 6 divided by 49 + 9\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "22 - 26i divided by 58\r<BR>\n" );
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document.write( "divide both numerator and denominator by 2 to get:\r<BR>\n" );
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document.write( "11 - 13i divided by 29\r<BR>\n" );
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document.write( "that should be your answer if i did it right.\r<BR>\n" );
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document.write( "here's a decent reference that pretty much tells you the same thing.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.sparknotes.com/math/algebra2/complexnumbers/section3.rhtml\" target = \"_blank\">http://www.sparknotes.com/math/algebra2/complexnumbers/section3.rhtml</a>\r<BR>\n" );
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document.write( "you remove the complex number from the denominator by multiplying the numerator and the denominator by the conjugate of the denominator.\r<BR>\n" );
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document.write( "if the denominator is a + bi, then the conjugate is a - bi\r<BR>\n" );
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document.write( "if the denominator is a - bi, then the conjugate is a + bi\r<BR>\n" );
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document.write( "both the numerator and the denominator have to be multiplied by the same factor in order to preserve the integrity of the expression.\r<BR>\n" );
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document.write( "2/4 * 2/2 = 4/8\r<BR>\n" );
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document.write( "the integrity of the expression is maintained because 4/8 is equivalent to 2/4.\r<BR>\n" );
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document.write( "between what i told you and the reference, you should get the idea.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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