document.write( "Question 1208359: Simplify sqrt{4x^3•9y^5}.\r
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Algebra.Com's Answer #846740 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "This problem is a TRAP, and both tutors Edwin and @math_tutor2020 fell into it.\r
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document.write( "To solve the problem correctly, we should reveal a context accurately.\r\n" );
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document.write( "The domain, i.e. the area where this expression is defined, is the set of points (x,y)\r\n" );
document.write( "in the 1st quadrant or in the 3rd quadrant.  It means that the domain is the union of both quadrants.\r\n" );
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document.write( "In other words x >= 0, y >= 0 are allowed under the square root; or x <= 0, y <= 0 simultaneously.\r\n" );
document.write( "But x and y of different signs are not allowed.\r\n" );
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document.write( "Second note is that when square root is considered as a function or as an expression,\r\n" );
document.write( "by a commonly accepted agreement, it is considered as non-negative value or expression.\r\n" );
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document.write( "    Again, it is the common agreement for square root, when it is considered \r\n" );
document.write( "    as an expression of variables, that its output values are non-negative.\r\n" );
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document.write( "Next, the first move is to write  \"sqrt%284x%5E3%2A9y%5E5%29\" = \"2%2A3%2Ax%2Ay%5E2%2Asqrt%28xy%29\".    (1)\r\n" );
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document.write( "But this expression is only half-right.  On the other half it is WRONG.\r\n" );
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document.write( "It is wrong in 3rd quadrant.  Numerical values under the square root are non-negative there. \r\n" );
document.write( "It is good, since we can extract the square root then.\r\n" );
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document.write( "But the values of the simplified expression (1) will be negative in 3rd quadrant, \r\n" );
document.write( "which is not consistent with the agreement mentioned above.\r\n" );
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document.write( "Therefore, we should write (1) in MODIFIED form\r\n" );
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document.write( "    \"sqrt%284x%5E3%2A9y%5E5%29\" = \"2%2A3%2Aabs%28x%29%2Ay%5E2%2Asqrt%28xy%29\" = \"6%2Aabs%28x%29%2Ay%5E2%2Asqrt%28xy%29\",   (2)\r\n" );
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document.write( "placing x under the absolute value sign.\r\n" );
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document.write( "In this form (2),  it is consistent with (1) in the first quadrant and provides non-negative output\r\n" );
document.write( "in the third quadrant.\r\n" );
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document.write( "ANSWER.  \"sqrt%284x%5E3%2A9y%5E5%29\"}  is  \"6%2Aabs%28x%29%2Ay%5E2%2Asqrt%28xy%29\"  in the 1st and 3rd quadrants.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved. The correct answer is expression (2).\r
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\n" ); document.write( "\n" ); document.write( "For your info and for info of the tutors, practically all without exceptions fall into this trap,
\n" ); document.write( "if they are unfamiliar with this logic/reasoning for advance, until somebody explains a right way reasoning.\r
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\n" ); document.write( "\n" ); document.write( "So, to get the right answer to this problem and to many other similar problems,
\n" ); document.write( "you need accurately determine the domain, accurately reveal the context and accurately
\n" ); document.write( "adopt the output expression in order for to have positive values of the final square root expression over the entire domain.\r
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\n" ); document.write( "\n" ); document.write( "It is the KEY to solving such problems.\r
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\n" ); document.write( "\n" ); document.write( "The key point to understand is that when square root of a non-negative number is considered,
\n" ); document.write( "it has two values - one positive and one negative, opposite to the positive value.\r
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\n" ); document.write( "\n" ); document.write( "On contrary, when square root of a function or of an expression is considered,
\n" ); document.write( "the common agreement is that in this situation the output (the result of applying square root) must be non-negative.\r
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\n" ); document.write( "\n" ); document.write( "It requires special cares in the solution.\r
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\n" ); document.write( "\n" ); document.write( "I think that 99% (or 99.99%) of contemporary people/students/teachers don't know this wisdom.\r
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\n" ); document.write( "\n" ); document.write( "But it is how my teachers taught us in my high school years.\r
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\n" ); document.write( "\n" ); document.write( "By the way, at the time, when I entered the university, we were oblige to pass
\n" ); document.write( "entering exams; two of the exams were in Math, written and oral.\r
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\n" ); document.write( "\n" ); document.write( "The competition for admission was 10:1; the participants were the best mathematical minds
\n" ); document.write( "of my generation from the entire country, and those who did not know (did not have) this wisdom/understanding
\n" ); document.write( "or other similar wisdoms/knowledge/technique/skills, had no chance to pass.\r
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\n" ); document.write( "\n" ); document.write( "And the last argument, to make you smile and to convince you in full.\r
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\n" ); document.write( "\n" ); document.write( "I asked my Google browser \" simplify \"sqrt%28x%5E3%2Ay%5E5%29\" \".\r
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\n" ); document.write( "\n" ); document.write( "It ran its AI, which generated the solution and the answer \"abs%28x%29%2Ay%5E2%2Asqrt%28xy%29\" ,
\n" ); document.write( "in precise accordance with my logic.\r
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\n" ); document.write( "\n" ); document.write( "The link is\r
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\n" ); document.write( "\n" ); document.write( "https://www.google.com/search?q=simplify+sqrt%28x%5E3*y%5E5%29&sca_esv=d744b37d0a57b190&sca_upv=1&rlz=1C1CHBF_enUS1071US1071&sxsrf=ADLYWIJQbRLqpb3gj7WgO1QlYzC1Q4VUsg%3A1727563466077&ei=yob4ZtKwBI_n0PEP1paqoQc&ved=0ahUKEwjSveO12-aIAxWPMzQIHVaLKnQQ4dUDCA8&uact=5&oq=simplify+sqrt%28x%5E3*y%5E5%29&gs_lp=Egxnd3Mtd2l6LXNlcnAiFnNpbXBsaWZ5IHNxcnQoeF4zKnleNSkyBRAhGJ8FMgUQIRifBTIFECEYnwUyBRAhGJ8FMgUQIRifBTIFECEYnwUyBRAhGJ8FMgUQIRifBUizOlDvDVj9MHABeAGQAQCYAWqgAcEEqgEDNS4xuAEDyAEA-AEBmAIHoALiBMICChAAGLADGNYEGEfCAgYQABgWGB7CAgsQABiABBiGAxiKBcICCBAAGIAEGKIEmAMAiAYBkAYIkgcDNC4zoAeuLg&sclient=gws-wiz-serp\r
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