document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=365979>Question 365979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is the last digit in the example 3 raised to the power of 2009?? and what is the solution?? is there a shortcut method to find the answers of the exponent<BR>\n" );
document.write( "??]\r<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 #260817 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=260817\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> yes there is.\r<BR>\n" );
document.write( "\n" );
document.write( "3^0 = 1 last digit is 1<BR>\n" );
document.write( "3^1 = 3 last digit is 3<BR>\n" );
document.write( "3^2 = 9 last digit is 9<BR>\n" );
document.write( "3^3 = 27 last digit is 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3^4 = 81 last digit is 1<BR>\n" );
document.write( "3^5 = 243 last digit is 3<BR>\n" );
document.write( "3^6 = 729 last digit is 9<BR>\n" );
document.write( "3^7 = 2187 last digit is 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3^8 = 6561 last digit is 1<BR>\n" );
document.write( "3^9 = 19683 last digit is 3<BR>\n" );
document.write( "3^10 = 59049 last digit is 9<BR>\n" );
document.write( "3^11 = 177147 last digit is 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "3^12 = 531441 last digit is 1<BR>\n" );
document.write( "3^13 = 1594323 last digit is 3<BR>\n" );
document.write( "3^14 = 4782969 last digit is 9<BR>\n" );
document.write( "3^15 = 14378907 last digit is 7\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "etc.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the last digits repeat every 4 exponents.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The remainder of dividing the exponent by 4 gives us the equivalent base exponent used.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let R(e/4) be the remainder of dividing the exponent by 4.1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Examples:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the exponent is 4, then the equivalent base exponent is R(4/4)) = 0.<BR>\n" );
document.write( "if the exponent is 5, then the equivalent base exponent is R(5/4)) = 1.<BR>\n" );
document.write( "if the exponent is 6, then the equivalent base exponent is R(6/4)) = 2.<BR>\n" );
document.write( "if the exponent is 7, then the equivalent base exponent is R(7/4)) = 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the exponent is 8, then the equivalent base exponent is R(8/4)) = 0.<BR>\n" );
document.write( "if the exponent is 9, then the equivalent base exponent is R(9/4)) = 1.<BR>\n" );
document.write( "if the exponent is 10, then the equivalent base exponent is R(10/4)) = 2.<BR>\n" );
document.write( "if the exponent is 11, then the equivalent base exponent is R(11/4)) = 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the exponent is 12, then the equivalent base exponent is R(12/4)) = 0.<BR>\n" );
document.write( "if the exponent is 13, then the equivalent base exponent is R(13/4)) = 1.<BR>\n" );
document.write( "if the exponent is 14, then the equivalent base exponent is R(14/4)) = 2.<BR>\n" );
document.write( "if the exponent is 15, then the equivalent base exponent is R(15/4)) = 3.\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( "if the base exponent is 0, then the last digit is 1.<BR>\n" );
document.write( "if the base exponent is 1, then the last digit is 3.<BR>\n" );
document.write( "if the base exponent is 2, then the last digit = 9.<BR>\n" );
document.write( "if the base exponent is 3, then the last digit is 7.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using this formula:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if the exponent is 2009, then the equivalent base exponent is R(2009/4)) = 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What you get is 2009/4 = 502.25\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( ".25 * 4 = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the remainder is 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the last digit is 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );