document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=85028>Question 85028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation:<BR>\n" );
document.write( "the quantity of x plus the square root of y..cubed<BR>\n" );
document.write( "i'm not sure how to write it on the computer--<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Please+help+me+solve+this+equation%3A%0D%0Athe+quantity+of+x+plus+the+square+root+of+y..cubed%0D%0Ai%27m+not+sure+how+to+write+it+on+the+computer--%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >(x + square root y)^3<BR>\n" );
document.write( "i know that i should separate it all out, but how do you foil a cubed equation? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #61277 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=61277\">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=%28x+%2B+sqrt%28y%29%29%5E3 ALIGN=MIDDLE  ALT=\"%28x+%2B+sqrt%28y%29%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Two lessons in here.  This will probably be better to work with if you replace the square<BR>\n" );
document.write( "root sign with its equivalent exponent. Remember that raising something to the (1/2) power<BR>\n" );
document.write( "is the same as taking the square root of it. So let's replace the square root of y with the<BR>\n" );
document.write( "quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"y%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This makes the problem become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%2B+y%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"x+%2B+y%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next let's think about what is going on when you use the FOIL method.  The term FOIL is<BR>\n" );
document.write( "just a memory jogger to help you remember what you can do to multiply two quantities<BR>\n" );
document.write( "that have two terms in each quantity. Let's take a little time to think about what you<BR>\n" );
document.write( "are doing when you \"FOIL\".  Suppose you have the two terms:<BR>\n" );
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document.write( "(a + b)*(c + d)<BR>\n" );
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document.write( "If you FOIL them you get (a*c)+(a*d)+(b*c)+(b*d).  Now I'm going to ask you to look at<BR>\n" );
document.write( "what you did. You took the first term in the first set of parentheses and you used it<BR>\n" );
document.write( "to multiply both terms in the second set. Then you took the second term in the first set<BR>\n" );
document.write( "of parentheses and you used it to multiply both terms in the second set. No magic there,<BR>\n" );
document.write( "but you lose sight of what you are doing by saying Firsts, Outsides, Insides, Lasts.<BR>\n" );
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document.write( "Now I'm going to propose something very similar to FOIL, but one set of parentheses<BR>\n" );
document.write( "will contain three terms. Suppose you have:<BR>\n" );
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document.write( "(a + b)*(c + d + e)<BR>\n" );
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document.write( "How do you do that? FOIL doesn't apply does it?  Well, you just use a process similar to<BR>\n" );
document.write( "the one I suggested in the first case. Take the first term in the first set of parentheses<BR>\n" );
document.write( "(that is the term \"a\") and you multiply it by all three terms in the second set so that you<BR>\n" );
document.write( "get ac + ad + ae. Then you take the second term in the first set (that term is \"b\") and<BR>\n" );
document.write( "you multiply it by all three terms in the second set and you get bc + bd + be. Then you<BR>\n" );
document.write( "combine the two sets of answers to get ac + ad + ae + bc + bd + be. And you can use the<BR>\n" );
document.write( "same process for other problems. Take the terms one at a time from the first set of parentheses<BR>\n" );
document.write( "and multiply them by each of the terms in the second set. When you are finished with all<BR>\n" );
document.write( "the terms from the first set of parentheses, add all the answers together to get the whole<BR>\n" );
document.write( "product.<BR>\n" );
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document.write( "To me, it's easier to remember that method that FOIL, and it works for all sorts of products ...<BR>\n" );
document.write( "even ones such as (a + b + c + d)*(e + f + g + h + i)<BR>\n" );
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document.write( "For this example you would take a and multiply it by all 5 terms in the second set, then<BR>\n" );
document.write( "take b and multiply it by all 5, then take c and do the same, then take d and also do<BR>\n" );
document.write( "the same.  When you get through with all those just add them up and you have the answer.<BR>\n" );
document.write( "Try doing that with FOIL ...<BR>\n" );
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document.write( "OK. Back to your problem. We said that we would substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"y%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for the square root<BR>\n" );
document.write( "of y and the problem would, therefore, become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+y%5E%281%2F2%29%29%5E3 ALIGN=MIDDLE  ALT=\"%28x+%2B+y%5E%281%2F2%29%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "I think you were on the right track thinking that you could split this into:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+y%5E%281%2F2%29%29%2A%28x+%2B+y%5E%281%2F2%29%29%2A%28x+%2B+y%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+y%5E%281%2F2%29%29%2A%28x+%2B+y%5E%281%2F2%29%29%2A%28x+%2B+y%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which would be the same as cubing it.  Then you would take the first two terms and FOIL<BR>\n" );
document.write( "them to get the square of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28x%2By%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and then multiply that by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+y%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+y%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "to get the answer.<BR>\n" );
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document.write( "So let's multiply the first two terms: <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+y%5E%281%2F2%29%29%2A%28x+%2B+y%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+y%5E%281%2F2%29%29%2A%28x+%2B+y%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiplying the x times all the terms in the second set of parentheses and then doing the same<BR>\n" );
document.write( "for multiplying the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"y%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > times the terms in the second set results in:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+xy%5E%281%2F2%29+%2B+xy%5E%281%2F2%29+%2B+y ALIGN=MIDDLE  ALT=\"x%5E2+%2B+xy%5E%281%2F2%29+%2B+xy%5E%281%2F2%29+%2B+y\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F2%29%2Ay%5E%281%2F2%29+=+y%5E%281%2F2%2B1%2F2%29+=+y+%5E1+=+y ALIGN=MIDDLE  ALT=\"y%5E%281%2F2%29%2Ay%5E%281%2F2%29+=+y%5E%281%2F2%2B1%2F2%29+=+y+%5E1+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >. And you can combine the two<BR>\n" );
document.write( "terms in the middle.  As a result the product of the first two terms is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B2xy%5E%281%2F2%29+%2B+y+ ALIGN=MIDDLE  ALT=\"x%5E2+%2B2xy%5E%281%2F2%29+%2B+y+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now you need to multiply that again by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+y%5E%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+y%5E%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This product is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+%2B+y%5E%281%2F2%29%29%2A%28x%5E2+%2B+2xy%5E%281%2F2%29+%2B+y%29 ALIGN=MIDDLE  ALT=\"%28x+%2B+y%5E%281%2F2%29%29%2A%28x%5E2+%2B+2xy%5E%281%2F2%29+%2B+y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Use the new rule by multiplying x times every term in the second set and then multiplying<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"y%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by everything in the second set and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+2x%5E2y%5E%281%2F2%29+%2B+xy+%2B+x%5E2y%5E%281%2F2%29%2B+2xy+%2B+y%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+2x%5E2y%5E%281%2F2%29+%2B+xy+%2B+x%5E2y%5E%281%2F2%29%2B+2xy+%2B+y%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%281%2F2%29%2Ay+=+y%5E%281%2F2%29%2Ay%5E%282%2F2%29+=+y%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"y%5E%281%2F2%29%2Ay+=+y%5E%281%2F2%29%2Ay%5E%282%2F2%29+=+y%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note also that you have some terms<BR>\n" );
document.write( "(a couple of sets) that can be combined. After like terms are combined the answer is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+%2B+3x%5E2y%5E%281%2F2%29+%2B+3xy+%2B+y%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"x%5E3+%2B+3x%5E2y%5E%281%2F2%29+%2B+3xy+%2B+y%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That's the answer. Hope this helps to clarify the problem for you. I noticed that in this<BR>\n" );
document.write( "problem the formula translator clips a little bit off the tops of the numerators in the<BR>\n" );
document.write( "exponents.  Look carefully. In some cases the exponents are 1/2 and in others 3/2, but<BR>\n" );
document.write( "you should be able to tell the difference. Sorry about that. </SPAN>\n" );
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