document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197176>Question 1197176</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is the following simplification correct? Why or why not?  Use complete sentences to explain your answer.\r<BR>\n" );
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document.write( "2^5 * 2^7 = 4^12\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830327 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=830327\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The general rule is<BR>\n" );
document.write( "a^b*a^c = a^(b+c)\r<BR>\n" );
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document.write( "The bases are the same. In this case, the base is 'a'<BR>\n" );
document.write( "We add the exponents b and c to arrive at a single exponential expression on the right hand side.\r<BR>\n" );
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document.write( "For this problem<BR>\n" );
document.write( "a = 2<BR>\n" );
document.write( "b = 5<BR>\n" );
document.write( "c = 7<BR>\n" );
document.write( "So it should be<BR>\n" );
document.write( "a^b*a^c = a^(b+c)<BR>\n" );
document.write( "2^5*2^7 = 2^(5+7) = 2^12\r<BR>\n" );
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document.write( "Therefore the equation 2^5*2^7 = 4^12 is false.<BR>\n" );
document.write( "The '4' should be a 2.\r<BR>\n" );
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document.write( "We can use a calculator to evaluate each expression<BR>\n" );
document.write( "2^5*2^7 = 4,096<BR>\n" );
document.write( "4^12 = 16,777,216<BR>\n" );
document.write( "2^12 = 4,096<BR>\n" );
document.write( "This helps show 2^5*2^7 = 2^12 is a true statement.\r<BR>\n" );
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document.write( "-------------------------------------------------------\r<BR>\n" );
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document.write( "If you're wondering how the rule a^b*a^c = a^(b+c) works, then let's break down what 2^5 and 2^7 mean.\r<BR>\n" );
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document.write( "2^5 means \"multiply 5 copies of the base 2\"<BR>\n" );
document.write( "2^7 means \"multiply 7 copies of the base 2\"\r<BR>\n" );
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document.write( "We can write it out long hand like this<BR>\n" );
document.write( "2^5 = (2*2*2)*2*2<BR>\n" );
document.write( "2^7 = (2*2*2)*(2*2*2)*2<BR>\n" );
document.write( "The parenthesis are useful to group terms, or we might get lost in a sea of '2's. \r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "2^5*2^7 = [ 2^5 ] * [ 2^7 ]<BR>\n" );
document.write( "2^5*2^7 = [ (2*2*2)*2*2 ] * [ (2*2*2)*(2*2*2)*2 ]<BR>\n" );
document.write( "2^5*2^7 = (2*2*2)*(2*2*2)*(2*2*2)*(2*2*2)<BR>\n" );
document.write( "2^5*2^7 = 2^12\r<BR>\n" );
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document.write( "In other words,<BR>\n" );
document.write( "2^5*2^7 = [  2^5  ] * [  2^7 ]<BR>\n" );
document.write( "2^5*2^7 = [  5 copies of '2' multiplied  ] * [  7 copies of '2' multiplied ]<BR>\n" );
document.write( "2^5*2^7 = (5+7) copies of '2' multiplied<BR>\n" );
document.write( "2^5*2^7 = 12 copies of '2' multiplied<BR>\n" );
document.write( "2^5*2^7 = 2^12\r<BR>\n" );
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document.write( "This is no means a full formal proof of the rule a^b*a^c = a^(b+c), but it hopefully helps illustrate why the rule works.<BR>\n" );
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