document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=936945>Question 936945</A>: <I><SPAN STYLE=\"word-break: break-all;\"> We use the Greek letter iota, i, to represent the square root of -1 ( ). We call i an imaginary unit. Any number that is a product of a real number and the imaginary unit i is called an imaginary number. Complex numbers are numbers that consist of a real part and an imaginary part, and they are generally expressed as a + bi. In this task, you will find a pattern relating to the powers of i and also discover some identities involving complex numbers.\r<BR>\n" );
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document.write( "a.	Use the identity i2 = -1 to compute the powers of i and complete the table.\r<BR>\n" );
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document.write( "Type your response here:<BR>\n" );
document.write( "Power of i	Result<BR>\n" );
document.write( "i^3	<BR>\n" );
document.write( "i^4	<BR>\n" );
document.write( "i^5	<BR>\n" );
document.write( "i^6	<BR>\n" );
document.write( "i^7	<BR>\n" );
document.write( "i^4n	<BR>\n" );
document.write( "i^4n+1	<BR>\n" );
document.write( "i^4n+2	<BR>\n" );
document.write( "i^4n+3	\r<BR>\n" );
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document.write( "b. Examine the pattern in the powers of i you wrote in the table, and create a rule for finding the value of large powers of i. Justify your answer.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #570396 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=570396\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. Use the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E2+=+-1%29%29%29+to+compute+the+powers+of+%7B%7B%7Bi ALIGN=MIDDLE  ALT=\"i%5E2+=+-1%29%29%29+to+compute+the+powers+of+%7B%7B%7Bi\"   DISABLED_style= \"max-width:90%; height:auto;\" > and complete the table.<BR>\n" );
document.write( "Type your response here:<BR>\n" );
document.write( "Power of i Result<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i%5E3=i%5E2%2Ai=%28-1%29%2Ai=-i ALIGN=MIDDLE  ALT=\"i%5E3=i%5E2%2Ai=%28-1%29%2Ai=-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%5E2%2Ai%5E2=%28-1%29%2A%28-1%29=1 ALIGN=MIDDLE  ALT=\"i%5E4=i%5E2%2Ai%5E2=%28-1%29%2A%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%5E2%2Ai%5E2%2Ai=%28-1%29%2A%28-1%29%2Ai=i ALIGN=MIDDLE  ALT=\"i%5E5=i%5E2%2Ai%5E2%2Ai=%28-1%29%2A%28-1%29%2Ai=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%5E2%2Ai%5E2%2Ai%5E2=%28-1%29%2A%28-1%29%2A%28-1%29=-1 ALIGN=MIDDLE  ALT=\"i%5E6=i%5E2%2Ai%5E2%2Ai%5E2=%28-1%29%2A%28-1%29%2A%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%5E7=i%5E2%2Ai%5E2%2Ai%5E2%2Ai=%28-1%29%2A%28-1%29%2A%28-1%29%2Ai=-i ALIGN=MIDDLE  ALT=\"i%5E7=i%5E2%2Ai%5E2%2Ai%5E2%2Ai=%28-1%29%2A%28-1%29%2A%28-1%29%2Ai=-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%5E4n=%28i%5E4%29%5En=%28i%5E2%2Ai%5E2%29%5En=%28%28-1%29%2A%28-1%29%29%5En=1%5En=1 ALIGN=MIDDLE  ALT=\"i%5E4n=%28i%5E4%29%5En=%28i%5E2%2Ai%5E2%29%5En=%28%28-1%29%2A%28-1%29%29%5En=1%5En=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%5E%284n%2B1%29=%28i%5E4%29%5En%2Ai=1%2Ai=i ALIGN=MIDDLE  ALT=\"i%5E%284n%2B1%29=%28i%5E4%29%5En%2Ai=1%2Ai=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%5E%284n%2B2%29=%28i%5E4%29%5En%2Ai%5E2=1%2A%28-1%29=-1 ALIGN=MIDDLE  ALT=\"i%5E%284n%2B2%29=%28i%5E4%29%5En%2Ai%5E2=1%2A%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%5E%284n%2B3%29=%28i%5E4%29%5En%2Ai%5E2%2Ai=1%2A%28-1%29%2Ai=-i ALIGN=MIDDLE  ALT=\"i%5E%284n%2B3%29=%28i%5E4%29%5En%2Ai%5E2%2Ai=1%2A%28-1%29%2Ai=-i\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b. Examine the pattern in the powers of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" > you wrote in the table, and create a rule for finding the value of large powers of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=i ALIGN=MIDDLE  ALT=\"i\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Justify your answer. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5En ALIGN=MIDDLE  ALT=\"b%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the product of multiplying <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > bases:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5En=b%2Ab%2Ab ALIGN=MIDDLE  ALT=\"b%5En=b%2Ab%2Ab\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+ ALIGN=MIDDLE  ALT=\"b+\"   DISABLED_style= \"max-width:90%; height:auto;\" > and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is multiplied by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+n ALIGN=MIDDLE  ALT=\"+n\"   DISABLED_style= \"max-width:90%; height:auto;\" >  times\r<BR>\n" );
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document.write( "so,we can apply same rule if  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=+i%5E2 ALIGN=MIDDLE  ALT=\"b=+i%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we will have \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28i%5E2%29%5En=i%5E%282n%29=i%5E2%2Ai%5E2 ALIGN=MIDDLE  ALT=\"%28i%5E2%29%5En=i%5E%282n%29=i%5E2%2Ai%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......<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;\" >......   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > times\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );