document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=397013>Question 397013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This one is really bugging me:\r<BR>\n" );
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document.write( "I understand that 2(2^n-1) (that is, two times two to the n-1 power) is equal to 2^n since the rule is \"add the exponents if the bases are equal\"). But why can't I compute it this way: 2(2^n-1) = 4^n-1 which could be rewritten as (2^2)^n-1 which, following the rule \"multiply exponents when you have something to a power taken to another power\" would then distribute as 2^2(n-1) = 2^2n-2. But this is a completely different answer!\r<BR>\n" );
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document.write( "This doesn't make any since so surely I am making an obvious mistake. But given, for example 2(x^n+1) don't get 2x^n+1???\r<BR>\n" );
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document.write( "Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #281477 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=281477\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is what's going on (skip this part if you know this already)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282%5E%28n-1%29%29 ALIGN=MIDDLE  ALT=\"2%282%5E%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E1%282%5E%28n-1%29%29 ALIGN=MIDDLE  ALT=\"2%5E1%282%5E%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E1 ALIGN=MIDDLE  ALT=\"2%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we're going to use the identity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28y%29%2Ax%5E%28z%29=x%5E%28y%2Bz%29 ALIGN=MIDDLE  ALT=\"x%5E%28y%29%2Ax%5E%28z%29=x%5E%28y%2Bz%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%281%2Bn-1%29 ALIGN=MIDDLE  ALT=\"2%5E%281%2Bn-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply (by adding exponents) using the identity given above\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5En ALIGN=MIDDLE  ALT=\"2%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282%5E%28n-1%29%29=2%5En ALIGN=MIDDLE  ALT=\"2%282%5E%28n-1%29%29=2%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "----------------------------------------------------\r<BR>\n" );
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document.write( "Now your question is why isn't <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282%5E%28n-1%29%29 ALIGN=MIDDLE  ALT=\"2%282%5E%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"4%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. The simple reason why not is because you can't simply multiply the bases. Remember that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E1 ALIGN=MIDDLE  ALT=\"2%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "For example, we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3=8 ALIGN=MIDDLE  ALT=\"2%5E3=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E3=2%2A8=16 ALIGN=MIDDLE  ALT=\"2%2A2%5E3=2%2A8=16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But if we multiply the bases, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%5E3=4%5E3=64 ALIGN=MIDDLE  ALT=\"2%2A2%5E3=4%5E3=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is NOT true.\r<BR>\n" );
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document.write( "Also, you CANNOT rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282%5E%28n-1%29%29 ALIGN=MIDDLE  ALT=\"2%282%5E%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E2%29%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"%282%5E2%29%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the two are completely different. If you're skeptical about that, plug various values of 'n' into each expression and you'll see different results.\r<BR>\n" );
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document.write( "So unfortunately, your line of reasoning was based on a false assumption which lead to that contradiction.\r<BR>\n" );
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document.write( "If you need more help, email me at <a  rel=nofollow HREF=\"mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help\">jim_thompson5910@hotmail.com</a>\r<BR>\n" );
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document.write( "Also, feel free to check out my <a  rel=nofollow HREF=\"http://www.freewebs.com/jimthompson5910/home.html\">tutoring website</a>\r<BR>\n" );
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document.write( "Jim </SPAN>\n" );
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