document.write( "Question 1206857: Simplify cube root of (2 + √5) \n" ); document.write( "
Algebra.Com's Answer #844546 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We seek to find a simpler form \"root%283%2C2%2Bsqrt%285%29%29\"} \r\n" );
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document.write( "We try to find a simpler expression of the same form as what's \r\n" );
document.write( "under the radical, namely a surd of the form \"a%2Bb%2Asqrt%285%29\", where \r\n" );
document.write( "a and b are rational numbers.\r\n" );
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document.write( "If this is possible, then\r\n" );
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document.write( "\"root%283%2C2%2Bsqrt%285%29%29\"\"%22%22=%22%22\"\"a%2Bb%2Asqrt%285%29\"\r\n" );
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document.write( "Cubing both sides:\r\n" );
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document.write( "\"%28root%283%2C2%2Bsqrt%285%29%29%29%5E3\"\"%22%22=%22%22\"\"%28a%2Bb%2Asqrt%285%29%29%5E3\"\r\n" );
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document.write( "\"2%2Bsqrt%285%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "\"2%2Bsqrt%285%29\"\"%22%22=%22%22\"\"a%5E3%2B3%2Aa%5E2%2Ab%2Asqrt%285%29%2B3%2Aa%2Ab%2A5%2Bb%5E3%285%2Asqrt%285%29%29\"\r\n" );
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document.write( "\"2%2Bsqrt%285%29\"\"%22%22=%22%22\"\"a%5E3%2B3%2Aa%5E2%2Ab%2Asqrt%285%29%2B15%2Aa%2Ab%5E2%2B5%2Ab%5E3%2Asqrt%285%29%29\"\r\n" );
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document.write( "The rational terms on the left must equal the rational terms on the right,\r\n" );
document.write( "and The irrational terms on the left must equal the irrational terms on the\r\n" );
document.write( "right, so we have this system\r\n" );
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document.write( "\"system%282=a%5E3%2B15ab%5E2%2C1=3a%5E2b%2B5b%5E3%29\"\r\n" );
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document.write( "Eliminate the constants by multiplying the 2nd equation by -2\r\n" );
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document.write( "\"system%282=a%5E3%2B15ab%5E2%2C-2=-6a%5E2b-10b%5E3%29\"\r\n" );
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document.write( "Adding the equations:\r\n" );
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document.write( "\"a%5E3-6a%5E2b%2B15ab%5E2-10b%5E3=0\"\r\n" );
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document.write( "Notice that the coefficients have sum 1-6+15-10 = 0.\r\n" );
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document.write( "That means a=b is a solution. Substituting a for b,\r\n" );
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document.write( "\"2=a%5E3%2B15ab%5E2\"\r\n" );
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document.write( "\"2=a%5E3%2B15a%5E3\"\r\n" );
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document.write( "\"2=16a%5E3\"\r\n" );
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document.write( "\"1=8a%5E3\"\r\n" );
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document.write( "\"1%2F8=a%5E3\"\r\n" );
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document.write( "Taking cube roots of both sides:\r\n" );
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document.write( "\"1%2F2=a\"\r\n" );
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document.write( "and since a=b, \r\n" );
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document.write( "\"a%2Bb%2Asqrt%285%29\"\r\n" );
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document.write( "becomes\r\n" );
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document.write( "\"1%2F2%2Bexpr%281%2F2%29sqrt%285%29\"\r\n" );
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document.write( "Edwin
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