document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1106522>Question 1106522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help!! find k if 5^1 x 5^2 x 5^3 x ... x 5^k = 5^171 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #721505 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=721505\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Rule: a^b*a^c = a^(b+c)\r<BR>\n" );
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document.write( "Using that rule, we can say that<BR>\n" );
document.write( "5^1*5^2 = 5^(1+2)<BR>\n" );
document.write( "and by extension<BR>\n" );
document.write( "5^1*5^2*5^3 = 5^(1+2)*5^3 = 5^(1+2+3)<BR>\n" );
document.write( "and so on\r<BR>\n" );
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document.write( "For the entire left side, we can condense<BR>\n" );
document.write( "5^1*5^2*5^3* ... * 5^k<BR>\n" );
document.write( "into<BR>\n" );
document.write( "5^(1+2+3+...+k)<BR>\n" );
document.write( "using that rule mentioned above\r<BR>\n" );
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document.write( "Therefore we have <BR>\n" );
document.write( "5^1*5^2*5^3* ... * 5^k = 5^171<BR>\n" );
document.write( "turn into<BR>\n" );
document.write( "5^(1+2+3+...+k) = 5^171\r<BR>\n" );
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document.write( "The bases are both equal to 5, so the exponents must be equal<BR>\n" );
document.write( "In other words,<BR>\n" );
document.write( "1+2+3+...+k = 171\r<BR>\n" );
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document.write( "The left side can be found using the formula <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle \sum_{m=1}^{k}m = 1+2+3+\ldots k = \frac{k(k+1)}{2}\">\r<BR>\n" );
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document.write( "Which is why <BR>\n" );
document.write( "1+2+3+...+k = 171<BR>\n" );
document.write( "becomes<BR>\n" );
document.write( "k*(k+1)/2 = 171\r<BR>\n" );
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document.write( "Now solve for k<BR>\n" );
document.write( "k*(k+1)/2 = 171<BR>\n" );
document.write( "k*(k+1) = 171*2<BR>\n" );
document.write( "k^2+k = 342<BR>\n" );
document.write( "k^2+k-342 = 0<BR>\n" );
document.write( "(k+19)(k-18) = 0<BR>\n" );
document.write( "k+19 = 0 or k-18 = 0<BR>\n" );
document.write( "k = -19 or k = 18\r<BR>\n" );
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document.write( "We only consider positive k values. So toss out k = -19<BR>\n" );
document.write( "The only solution is k = 18<BR>\n" );
document.write( "Note how 1+2+3+...+k = 1+2+3+...+18 = 171\r<BR>\n" );
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document.write( "If you want to see it more expanded out, then,<BR>\n" );
document.write( "1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18 = 171\r<BR>\n" );
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document.write( "Final Answer: k = 18<BR>\n" );
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