document.write( "Question 1196903: Solve for x.\r
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Algebra.Com's Answer #829955 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Your starting equation is \r\n" );
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document.write( "    \"sqrt%28x%29%5Ex\" = \"x%5E%28sqrt%28x%29%29\".       (1)\r\n" );
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document.write( "One solution is x= 1 (obvious).\r\n" );
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document.write( "Indeed,  left side is \"sqrt%281%29%5E1\" = \"1%5E1\" = 1.\r\n" );
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document.write( "         Right side is equal to 1 due to the same reason.\r\n" );
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document.write( "Now I will assume that x =/= 1 and will look for other solutions.\r\n" );
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document.write( "Take logarithm base 10 of both sides of equation (1).  You will get\r\n" );
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document.write( "    \"x%2Alog%28%28sqrt%28x%29%29%29\" = \"sqrt%28x%29%2Alog%28%28x%29%29\".\r\n" );
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document.write( "Divide both sides by log(x)  (we can do it safely, since we consider  x =/= 1.)\r\n" );
document.write( "You will get\r\n" );
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document.write( "    \"sqrt%28x%29%2Fx\" = \"log%28%28sqrt%28x%29%29%29%2Flog%28%28x%29%29\",\r\n" );
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document.write( "or, equivalently,\r\n" );
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document.write( "    \"sqrt%28x%29%2Fx\" = \"1%2F2\".\r\n" );
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document.write( "It implies\r\n" );
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document.write( "    \"2%2Asqrt%28x%29\" = x\r\n" );
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document.write( "and after squaring both sides,\r\n" );
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document.write( "    4x = \"x%5E2\".\r\n" );
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document.write( "It implies\r\n" );
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document.write( "    \"4x+-+x%5E2\" = 0\r\n" );
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document.write( "    x*(4-x) = 0,\r\n" );
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document.write( "    x = 0  or   x= 4.\r\n" );
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document.write( "You can check that the root x= 0 works in the original equation, since then each side is equal to 1.\r\n" );
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document.write( "So, the remaining solutions to the problem are  x= 0  and  x= 4.\r\n" );
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document.write( "ANSWER.  The given equation has three solutions  x= 0,  x= 1,  and x= 4.\r\n" );
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document.write( "                  To see that x= 0 is the solution, too, look at this plot below.\r\n" );
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document.write( "    \"+graph%28800%2C400%2C+-0.2%2C+1.2%2C+-1%2C+2%2C+sqrt%28x%29%5Ex%2C+x%5Esqrt%28x%29+%29+\"\r\n" );
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document.write( "                        Plots y = \"sqrt%28x%29%5Ex\"  (red)  and  y = \"x%5Esqrt%28x%29\"  (green)\r\n" );
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