document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207482>Question 1207482</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Write each expression in the standard form a+bi.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "How do I simplify i^n when n is a big number?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Two samples:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A. i^14\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "B. 2i^(8)(2 + i^4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845377 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=845377\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "When dealing with large numbers, try to look at small examples to see if a pattern can be formed.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We'll look at the first few powers of i<BR>\n" );
document.write( "i^0 = 1<BR>\n" );
document.write( "i^1 = i<BR>\n" );
document.write( "i^2 = ( sqrt(-1) )^2 = -1<BR>\n" );
document.write( "i^3 = i*i^2 = i*(-1) = -i<BR>\n" );
document.write( "i^4 = ( i^2 )^2 = (-1)^2 = 1<BR>\n" );
document.write( "i^5 = i*i^4 = i*1 = i<BR>\n" );
document.write( "Etc\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The summarized list would be<BR>\n" );
document.write( "i^0 = 1<BR>\n" );
document.write( "i^1 = i<BR>\n" );
document.write( "i^2 = -1<BR>\n" );
document.write( "i^3 = -i<BR>\n" );
document.write( "i^4 = 1<BR>\n" );
document.write( "i^5 = i<BR>\n" );
document.write( "Etc\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Be sure not to mix up the lowercase \"eye\" with the numeric digit \"one\". <BR>\n" );
document.write( "The two symbols unfortunately look similar at a quick glance.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Anyways, the pattern we found was: 1, i, -1, -i, 1, i, ...<BR>\n" );
document.write( "As you can see, once reaching 1, the pattern starts over again.<BR>\n" );
document.write( "Therefore, this sequence repeats itself every 4 elements.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Why is this useful? Because we can divide the large-ish exponent of 14 by 4 to determine the remainder.<BR>\n" );
document.write( "14/4 = 3 remainder 2<BR>\n" );
document.write( "The \"<font color=blue>remainder 2</font>\" leads us to i^<font color=blue>2</font> since it has exponent of 2.<BR>\n" );
document.write( "Ignore the quotient. We only care about the remainder.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i^14 = i^2 = <font color=red>-1</font> which is the <font color=red>answer to part A</font>.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another example:<BR>\n" );
document.write( "i^15 = i^3 = -i<BR>\n" );
document.write( "since 15/4 gives some quotient with remainder 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another example:<BR>\n" );
document.write( "i^98 = i^2 = -1<BR>\n" );
document.write( "since 98/4 yields remainder 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If in doubt, turn to something like WolframAlpha to help confirm the answer<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input/?i=i%5E14\">https://www.wolframalpha.com/input/?i=i%5E14</a><BR>\n" );
document.write( "I use this tool a lot. Another tool that I use often is GeoGebra.<BR>\n" );
document.write( "There are many other similar tools on the web which I encourage you to explore your favorite.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I'll leave part B for the student to handle. <BR>\n" );
document.write( "If you are still stuck, or have any questions, then please let me know.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );