document.write( "Question 1209546: 1) If x + y = 1 and x³ + y³ = 3,
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Algebra.Com's Answer #849404 by ikleyn(52776)\"\" \"About 
You can put this solution on YOUR website!
.
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        Part  (a)

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document.write( "Start from\r\n" );
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document.write( "    x + y = 1.\r\n" );
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document.write( "Raise it in degree 3.  You will get\r\n" );
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document.write( "    1 = (x+y)^3 = x^3 + 3x^2*y + 3x*y^2 + y^3 = (x^3 + y^3) + 3xy(x+y).\r\n" );
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document.write( "Replace here x^3 + y^3  by  3 (since it is given),  and replace  x+y  by  1.  You will get\r\n" );
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document.write( "    1 = 3 + 3xy ,\r\n" );
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document.write( "    3xy = -2,\r\n" );
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document.write( "     xy = \"-2%2F3\".\r\n" );
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document.write( "Now   \"1%2Fx%5E3\" + \"1%2Fy%5E3\" = \"%28x%5E3+%2B+y%5E3%29%2F%28x%5E3%2Ay%5E3%29\" = \"3%2F%28%28-2%2F3%29%5E3%29\" = \"%28-3%29%2A%283%5E3%2F2%5E3%29\" = \"-81%2F8\".\r\n" );
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document.write( "ANSWER.  At given condition,  \"1%2Fx%5E3\" + \"1%2Fy%5E3\" = \"-81%2F8\".\r\n" );
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        Part  (b)

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document.write( "Obviously, the value of x should be greater than 1.\r\n" );
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document.write( "One such value is easy to guess:  it is  x = 2^(1/4) = \"root%284%2C2%29\" = 1.189207115,  approximately.\r\n" );
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document.write( "Indeed,  (2^(1/4))^16 = \"2%5E4\" = 16  and,  therefore,\r\n" );
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document.write( "    x^(x^16) = (2^(1/4))^16 = \"2%5E4\" = 16,\r\n" );
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document.write( "so this value of x,  x = 2^(1/4)  is the solution to the given equation  x^(x¹⁶) = 16.\r\n" );
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document.write( "From the other side hand,  this function, f(x) = x^(x¹⁶)  is a monotonically increasing function \r\n" );
document.write( "in the domain x > 1, so, there is no other solution to the given equation in this domain.\r\n" );
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document.write( "        Thus, for positive real numbers, the only solution \r\n" );
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document.write( "            to the given equation  x^(x¹⁶) = 16  \r\n" );
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document.write( "        is  x = 2^(1/4) = \"root%284%2C2%29\" = 1.189207115  (approximately).\r\n" );
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document.write( "CHECK.  \"1.189207115%5E16\" = 16;  therefore,  x^(x¹⁶) = x^16 = 1.189207115^16 = 16,  which is correct.\r\n" );
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document.write( "Now,  \"x%5E20\" = (2^(1/4))^20 = 2^5 = 32.\r\n" );
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document.write( "ANSWER.  Under given condition,  \"x%5E20\" = 32.\r\n" );
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