document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=88347>Question 88347</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem:\r<BR>\n" );
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document.write( "EXPONENTS:<BR>\n" );
document.write( "Simplify.\r<BR>\n" );
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document.write( "#24. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E%284n-5%29%287%5E%282n%2B3%29%29%5E3=%287%5E%284-6n%29%29%5E4 ALIGN=MIDDLE  ALT=\"7%5E%284n-5%29%287%5E%282n%2B3%29%29%5E3=%287%5E%284-6n%29%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Please and 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 #64173 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=64173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%5E%284n-5%29%29%2A%287%5E%282n%2B3%29%29%5E3=%287%5E%284-6n%29%29%5E4 ALIGN=MIDDLE  ALT=\"%287%5E%284n-5%29%29%2A%287%5E%282n%2B3%29%29%5E3=%287%5E%284-6n%29%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "First, note that all the bases are 7, so the rules that depend on the bases being common can<BR>\n" );
document.write( "be used.<BR>\n" );
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document.write( "Next note that two of the terms have exponents. The applicable rule for this this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"%28x%5Ea%29%5Eb+=+x%5E%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "for the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%5E%282n+%2B+3%29%29%5E3 ALIGN=MIDDLE  ALT=\"%287%5E%282n+%2B+3%29%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > you have x = 7, a = 2n +3, and b = 3.  Substituting<BR>\n" );
document.write( "these values into the rule results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%5E%282n%2B3%29%29%5E3+=+7%5E%28%282n%2B3%29%2A3%29+=+7%5E%286n+%2B+9%29 ALIGN=MIDDLE  ALT=\"%287%5E%282n%2B3%29%29%5E3+=+7%5E%28%282n%2B3%29%2A3%29+=+7%5E%286n+%2B+9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Similarly, the other term of the original problem that also can be simplified using the<BR>\n" );
document.write( "power rule is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%5E%284-6n%29%29%5E4 ALIGN=MIDDLE  ALT=\"%287%5E%284-6n%29%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "For this term x = 7, a = (4 - 6n), and b = 3. Substituting these values into the power rule<BR>\n" );
document.write( "results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%287%5E%284-6n%29%29%5E4+=+7%5E%28%284-6n%29%2A4%29+=+7%5E%2816-24n%29 ALIGN=MIDDLE  ALT=\"%287%5E%284-6n%29%29%5E4+=+7%5E%28%284-6n%29%2A4%29+=+7%5E%2816-24n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substituting the two simplifications into the original given problem results in it becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E%284n-5%29%2A7%5E%286n+%2B+9%29=7%5E%2816-24n%29 ALIGN=MIDDLE  ALT=\"7%5E%284n-5%29%2A7%5E%286n+%2B+9%29=7%5E%2816-24n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now you can apply the rule for two terms of a common base, each term raised to an exponent, <BR>\n" );
document.write( "when the two terms are multiplied.  This rule states that the product is the common base<BR>\n" );
document.write( "raised to the sum of the exponents. In equation form this is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Ea+%2A+x%5Eb+=+x%5E%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"x%5Ea+%2A+x%5Eb+=+x%5E%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "On the left side of the equation we have developed there are two terms that are multiplied.<BR>\n" );
document.write( "In this product x = 7, a = 4n - 5, and b = 6n + 9. When you substitute these values into<BR>\n" );
document.write( "the left side of the equation you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E%284n-5%29%2A7%5E%286n+%2B+9%29+=+7%5E%28%284n-5%29%2B%286n+%2B+9%29%29+=+7%5E%2810n+%2B+4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This becomes the left side of the equation. The equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%5E%2810n+%2B+4%29=+7%5E%2816-24n%29 ALIGN=MIDDLE  ALT=\"7%5E%2810n+%2B+4%29=+7%5E%2816-24n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Note that since the bases on the left and right sides are both the same ... 7 ... for<BR>\n" );
document.write( "the equation to be true, the two exponents have to be the same. So therefore you can say:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10n+%2B+4+=+16+-+24n ALIGN=MIDDLE  ALT=\"10n+%2B+4+=+16+-+24n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "We can solve this equation by getting all the terms containing n on one side of the equation<BR>\n" );
document.write( "and all the terms that are just numbers on the other side.  Start by adding +24n to both sides.<BR>\n" );
document.write( "this makes the equation become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24n+%2B+10n+%2B+4+=+16+-+24n+%2B+24n ALIGN=MIDDLE  ALT=\"24n+%2B+10n+%2B+4+=+16+-+24n+%2B+24n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "combining the terms containing n reduces the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=34n+%2B+4+=+16 ALIGN=MIDDLE  ALT=\"34n+%2B+4+=+16\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next add -4 to both sides to eliminate the +4 on the left side. This makes the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=34n+=+12 ALIGN=MIDDLE  ALT=\"34n+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Finally, solve for n by dividing both sides by 34 to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+12%2F34+=+6%2F17 ALIGN=MIDDLE  ALT=\"n+=+12%2F34+=+6%2F17\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to see your way through the problem. </SPAN>\n" );
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