document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=26058>Question 26058</A>: <I><SPAN STYLE=\"word-break: break-all;\"> ok, i just DONT understand imaginary numbers in problems! i really need help! heres a problem: 7i^3 + 8i^12 +9i^17 -5i^16 = ?\r<BR>\n" );
document.write( "\n" );
document.write( "PLEASE HELP! i dont know what to do with the numbers in front...!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #14034 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=14034\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Imaginary numbers: In math, the lower-case letter i is used to stand for the square root of -1, thus: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29+=+i ALIGN=MIDDLE  ALT=\"sqrt%28-1%29+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "If you multiply i times i or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2 ALIGN=MIDDLE  ALT=\"i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > this is the same as multiplying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28-1%29sqrt%28-1%29+=+%28sqrt%28-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"sqrt%28-1%29sqrt%28-1%29+=+%28sqrt%28-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = -1 so you can see that<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;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Now if you keep doing this, that is, find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E3 ALIGN=MIDDLE  ALT=\"i%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E4 ALIGN=MIDDLE  ALT=\"i%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E5 ALIGN=MIDDLE  ALT=\"i%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you begin to see a definite pattern:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i+=+sqrt%28-1%29 ALIGN=MIDDLE  ALT=\"i+=+sqrt%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<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=i%5E3+=+i%28i%5E2%29 ALIGN=MIDDLE  ALT=\"i%5E3+=+i%28i%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%28-1%29+=+-i ALIGN=MIDDLE  ALT=\"i%28-1%29+=+-i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E4+=+i%28i%5E3%29 ALIGN=MIDDLE  ALT=\"i%5E4+=+i%28i%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%28-i%29+=+-i%5E2 ALIGN=MIDDLE  ALT=\"i%28-i%29+=+-i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28-1%29+=+1 ALIGN=MIDDLE  ALT=\"-%28-1%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E5+=+i%28i%5E4%29 ALIGN=MIDDLE  ALT=\"i%5E5+=+i%28i%5E4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%281%29+=+i ALIGN=MIDDLE  ALT=\"i%281%29+=+i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E6+=+i%28i%5E5%29 ALIGN=MIDDLE  ALT=\"i%5E6+=+i%28i%5E5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%28i%29+=+i%5E2 ALIGN=MIDDLE  ALT=\"i%28i%29+=+i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E7+=+i%28i%5E6%29 ALIGN=MIDDLE  ALT=\"i%5E7+=+i%28i%5E6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%28-1%29+=+-i ALIGN=MIDDLE  ALT=\"i%28-1%29+=+-i\"   DISABLED_style= \"max-width:90%; height:auto;\" > ...and so on.\r<BR>\n" );
document.write( "\n" );
document.write( "Do you see it? Sometimes it's helpful to constuct a table for the values of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5En ALIGN=MIDDLE  ALT=\"i%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > for different values of n:<BR>\n" );
document.write( "n----<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5En ALIGN=MIDDLE  ALT=\"i%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--------------<BR>\n" );
document.write( "1.....i<BR>\n" );
document.write( "2....-1<BR>\n" );
document.write( "3....-i<BR>\n" );
document.write( "4.....1<BR>\n" );
document.write( "5.....i<BR>\n" );
document.write( "6....-1<BR>\n" );
document.write( "7....-i<BR>\n" );
document.write( "8.....1<BR>\n" );
document.write( "9.....i<BR>\n" );
document.write( "10...-1<BR>\n" );
document.write( "11...-i<BR>\n" );
document.write( "12....1<BR>\n" );
document.write( "and so on<BR>\n" );
document.write( "As for the number in front of the i's in your problem, you just multiply by that number. For example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7i%5E3+=+7%28-i%29 ALIGN=MIDDLE  ALT=\"7i%5E3+=+7%28-i%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-7i ALIGN=MIDDLE  ALT=\"-7i\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8i%5E%2812%29+=+8%281%29 ALIGN=MIDDLE  ALT=\"8i%5E%2812%29+=+8%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );