document.write( "Question 1209466: Evaluate sqrt(y/x) where x and y are positive integers, and 0=x^5-(x^3y^3)-12393 \n" ); document.write( "

Algebra.Com's Answer #848890 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "Evaluate sqrt(y/x) where x and y are positive integers, and 0=x^5-(x^3y^3)-12393.
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document.write( "Your starting equation is\r\n" );
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document.write( "    x^5 - x^3*y^3 = 12393,      (1)\r\n" );
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document.write( "or\r\n" );
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document.write( "    x^3*(x^2 - y^3) = 12393.    (2)\r\n" );
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document.write( "Integer number 12393 has the primary decomposition  12393 = ,  \r\n" );
document.write( "so the last equation is\r\n" );
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document.write( "    x^3*(x^2 - y^3) = .   (3)\r\n" );
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document.write( "From it, it is clear that for x, y to be integer solutions to this equation, it is necessary that x be 1, or 3, or 3^2 = 9.\r\n" );
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document.write( "So, we should consider these three cases.\r\n" );
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document.write( "(a)  x = 1.  Then from equation (3)\r\n" );
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document.write( "     x^2 - y^3 = 3^6*17 = 12393,  y^3 = 1 - 12393 = -12392.\r\n" );
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document.write( "                                  But this number is not a positive perfect cube, so this way does not work.\r\n" );
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document.write( "(b)  x = 3.  Then from equation (3)\r\n" );
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document.write( "     x^2 - y^3 = 3^3*17 = 459,  y^3 = 9 - 459 = -450. \r\n" );
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document.write( "                                  But this number is not a positive perfect cube, so this way does not work.\r\n" );
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document.write( "(c)  x = 3^2 = 9.  Then from equation (3)\r\n" );
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document.write( "     x^2 - y^3 = 17,  y^3 = 81 - 17 = 64. \r\n" );
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document.write( "                                 This number, 64, is a positive perfect cube, so  y = 4.\r\n" );
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document.write( "Thus, the solution to the given equation in this pair of positive integer numbers  (x,y) = (3,4).\r\n" );
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document.write( "Then    is this irrational number  =  = 1.154700538  (rounded).    ANSWER\r\n" );
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