document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=638275>Question 638275</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me solve these problems. I am very confused and I need these problems completed in a few days. Any and all help will be greatly appreciated. Thank you.\r<BR>\n" );
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document.write( "key: sqrt = squareroot\r<BR>\n" );
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document.write( "1.   sqrt 27 + 2 sqrt 5 - sqrt 75\r<BR>\n" );
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document.write( "2. sqrt 5/4\r<BR>\n" );
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document.write( "3. sqrt 4/5\r<BR>\n" );
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document.write( "                                      Thank you sooo much for all of your help! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #402100 by </B><A HREF=/tutors/aboutme.mpl?userid=josh_jordan><B>josh_jordan(263)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josh_jordan><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/cgi-bin/show-face.mpl?user=josh_jordan><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=402100\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2827%29%2B2sqrt%285%29-sqrt%2875%29 ALIGN=MIDDLE  ALT=\"sqrt%2827%29%2B2sqrt%285%29-sqrt%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%2F4%29 ALIGN=MIDDLE  ALT=\"sqrt%285%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "3. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%284%2F5%29 ALIGN=MIDDLE  ALT=\"sqrt%284%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "1.  In order to solve this expression, we need to first simplify each root.  To do this, we need to look at the numbers inside each root and determine if they are the products of perfect squares, or at least one perfect square.  In other words, can any of these numbers be formed by multiplying two numbers together and one or both of those numbers are squares (numbers that are formed by multiplying two of the same numbers together)?  Let's look at the first root:  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2827%29 ALIGN=MIDDLE  ALT=\"sqrt%2827%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Are there any numbers that are squares that can be multiplied to another number to give you 27?  To answer this, let's look at all of the factors of 27 (aside from 1 and 27): 3 and 9. 3 is not a perfect square, but 9 is, because 3 x 3 = 9 (the same number can be multiplied by itself to give you 9).  So, 9 x 3 = 27.  Since 9 is a perfect square, we can take the square root of 9 and put it in the front of our square root, and keep the 3 inside the root since it is not a perfect square:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3sqrt%283%29 ALIGN=MIDDLE  ALT=\"3sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's look at the second root in this problem: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%285%29 ALIGN=MIDDLE  ALT=\"2sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  This cannot be broken down any further because the number inside the root,5, has no factors that are perfect squares.\r<BR>\n" );
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document.write( "Let's look at the third root in this problem: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2875%29 ALIGN=MIDDLE  ALT=\"sqrt%2875%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Let's look at the factors of 75 (aside from 1 and 75):  3, 5, 15, and 25.  3, 5, and 15 are not perfect squares, but 25 is.  So, 25 x 3 = 75.  We can take the square root of 25, and put it in front of our root:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5sqrt%283%29 ALIGN=MIDDLE  ALT=\"5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now, we will put all of these together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3sqrt%283%29%2B2sqrt%285%29%2B5sqrt%283%29 ALIGN=MIDDLE  ALT=\"3sqrt%283%29%2B2sqrt%285%29%2B5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "One of the laws of roots states that: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Asqrt%28m%29%2Bb%2Asqrt%28m%29=+%28a%2Bb%29sqrt%28m%29 ALIGN=MIDDLE  ALT=\"a%2Asqrt%28m%29%2Bb%2Asqrt%28m%29=+%28a%2Bb%29sqrt%28m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3sqrt%283%29+and+5sqrt%283%29 ALIGN=MIDDLE  ALT=\"3sqrt%283%29+and+5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > both have a root of 3, we can add the 3 and 5 together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3sqrt%283%29%2B5sqrt%283%29+=+8sqrt%283%29 ALIGN=MIDDLE  ALT=\"3sqrt%283%29%2B5sqrt%283%29+=+8sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%285%29+and+8sqrt%283%29 ALIGN=MIDDLE  ALT=\"2sqrt%285%29+and+8sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > have different roots, we cannot add them together.  Therefore, our final answer is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%285%29%2B8sqrt%283%29 ALIGN=MIDDLE  ALT=\"2sqrt%285%29%2B8sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  \r<BR>\n" );
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document.write( "2. Another law of roots is that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28m%2Fn%29+=+sqrt%28m%29%2Fsqrt%28n%29 ALIGN=MIDDLE  ALT=\"sqrt%28m%2Fn%29+=+sqrt%28m%29%2Fsqrt%28n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  So, to solve this problem, put the square root symbol around the numerator AND around the denominator:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2Fsqrt%284%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2Fsqrt%284%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Using the instructions for simplifying roots in answer 1, can you simplify the numerator?  No, because there are no perfect square numbers that can be multiplied by a different number to give you 5.  So, the numerator stays as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Can the denominator be simplified?  Yes, because the number 4 is a perfect square, because it's the product of 2 x 2.  So our denominator changes to 2.  Our final answer, therefore, is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29%2F2 ALIGN=MIDDLE  ALT=\"sqrt%285%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "3. This is alot like answer 2.  Almost.  Using the instructions in both answer 1 and answer 2, you will end up with:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"2%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "But, our final answer cannot have a root in the denominator.  We have to remove that square root.  To do so, we will use another law of roots, which states that: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28m%29%2Asqrt%28m%29=m ALIGN=MIDDLE  ALT=\"sqrt%28m%29%2Asqrt%28m%29=m\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So, to remove the root from the denominator, multiply both the denominator and numerator by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%285%29 ALIGN=MIDDLE  ALT=\"sqrt%285%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%2Asqrt%285%29%2F%28sqrt%285%29%2Asqrt%285%29%29 ALIGN=MIDDLE  ALT=\"2%2Asqrt%285%29%2F%28sqrt%285%29%2Asqrt%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "After multiplying, you will obtain your final answer, which is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%285%29%2F5 ALIGN=MIDDLE  ALT=\"2sqrt%285%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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