document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=917763>Question 917763</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Simplify <BR>\n" );
document.write( "(-i)^19 \r<BR>\n" );
document.write( "\n" );
document.write( "(-1)^(-21/2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #556811 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=556811\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> powers of i are calculated as follows:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\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\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this pattern repeats every 4 powers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i^4 = i^0 = 1<BR>\n" );
document.write( "i^5 = i^1 = i<BR>\n" );
document.write( "i^6 = i^2 = -1<BR>\n" );
document.write( "i^7 = i^3 = -i\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i^8 = i^0 = 1<BR>\n" );
document.write( "i^9 = i^1 = i<BR>\n" );
document.write( "i^10 = i^2 = -1<BR>\n" );
document.write( "i^11 = i^3 = -i\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "looking at the pattern, some smart person figured out that if you divide the exponent by 4, then you use the remainder to find the value of i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for example:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to find the value of i^4, divide 4 by 4 to get a remainder of 0 and so the value of i^4 is the same as the value of i^0 which is equal to 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to find the value of i^5, divide 5 by 4 to get a remainder of 1 and so the value of i^5 is the same as the value of i^1 which is equal to i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to find the value of i^10, divide 10 by 4 to get a remainder of 2 and so the value of i^10 is the same as the value of i^2 which is equal to -1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "etc.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "use of this formula will help you solve the higher exponent problems.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your first problem is to find the value of i^19.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "take 19 and divide it by 4 and you get a remainder of 3, so the value of i^19 = the value of i^3 which is equal to -i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now to problem 2:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(-1)^(-21/2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this can be solved as follows:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(-1)^(-21/2) is the same as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "((-1)^(1/2))^-21).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since (-1)^(1/2) is equal to sqrt(-1), this is the same as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "sqrt(-1)^-21.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since sqrt(-1) is equal to i, this is the same as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "i^-21.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since i^-21 is equal to 1/i^21, this is the same as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1/i^21.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since i^21 is equal to i^1 which is equal to i, this is the same as:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "1/i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can rationalize the denominator by multiplying the numerator and the denominator by i/i to get 1/i * i/i = i / i^2 which is equal to i / -1 which is equal to -i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that should be your solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for problem 1, the solution is -i<BR>\n" );
document.write( "for problem 2, the solution is also -i.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's an online calculator where you can check your work.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.mathsisfun.com/numbers/complex-number-calculator.html\" target = \"_blank\">http://www.mathsisfun.com/numbers/complex-number-calculator.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's a good tutorial on complex numbers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/complex.htm\" target= \"_blank\">http://www.purplemath.com/modules/complex.htm</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "here's the calculations i did in writing.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/102801.jpg\" alt=\"$$$\" </>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );