document.write( "Question 1208227: If x² = x + 1 and y² = y + 1.
\n" ); document.write( "Find x⁵ + y⁵
\n" ); document.write( "

Algebra.Com's Answer #846528 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\r\n" );
document.write( "There may be another way to work this using the sums of the powers and \r\n" );
document.write( "powers of the sums of x and y, here it is using the binomial expansion.\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2=x%2B1\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2-x-1=0\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%281+%2B-+sqrt%281%2B4%29%29%2F2+\"\r\n" );
document.write( "\r\n" );
document.write( "\"x+=+%281+%2B-+sqrt%285%29%29%2F2+\" \r\n" );
document.write( "\r\n" );
document.write( "\"x+=+1%2F2+%2B-+sqrt%285%29%2F2+\" \r\n" );
document.write( "\r\n" );
document.write( "y will be the same two values. Using the binomial theorem:\r\n" );
document.write( "\r\n" );
document.write( "\"%28a%2Bb%29%5E5\"\"%22%22=%22%22\"\"a%5E5%2B5a%5E4b%5E1%2B10a%5E3b%5E2%2B10a%5E2b%5E3%2B5ab%5E4%2Bb%5E5\"\r\n" );
document.write( "\r\n" );
document.write( "\"%281%2F2+%2B+sqrt%285%29%29%5E5+\"\"%22%22=%22%22\"\"%281%2F2%29%5E5%2B%28sqrt%285%29%2F2%29%5E5\"\"%22%22=%22%22\"\"%22%22=%22%22\"\r\n" );
document.write( "\"%22%22=%22%22\"\r\n" );
document.write( "\"%22%22=%22%22\"\r\n" );
document.write( "\r\n" );
document.write( "\"176%2F32%2B%2880sqrt%285%29%29%2F32\"\"%22%22=%22%22\"\"11%2F2%2B%285sqrt%285%29%29%2F2\"\r\n" );
document.write( "\r\n" );
document.write( "also, \"%281%2F2+-+sqrt%285%29%29%5E5+\"\"%22%22=%22%22\"\"11%2F2-%285sqrt%285%29%29%2F2\" \r\n" );
document.write( "because the expansion will be like the above except that all the terms with\r\n" );
document.write( "square roots will have - signs:\r\n" );
document.write( "\r\n" );
document.write( "So there are potentially 4 answers for x5 + y5,\r\n" );
document.write( "since x and y can be either one:\r\n" );
document.write( "\r\n" );
document.write( "(1) \"11%2F2%2B%285sqrt%285%29%29%2F2\"\"%22%22%2B%22%22\"\"11%2F2%2B%285sqrt%285%29%29%2F2\"\"%22%22=%22%22\"\"11%2B5sqrt%285%29\"\r\n" );
document.write( "\r\n" );
document.write( "(2) \"11%2F2%2B%285sqrt%285%29%29%2F2\"\"%22%22%2B%22%22\"\"11%2F2-%285sqrt%285%29%29%2F2\"\"%22%22=%22%22\"\"11\"\r\n" );
document.write( "\r\n" );
document.write( "(3) \"11%2F2-%285sqrt%285%29%29%2F2\"\"%22%22%2B%22%22\"\"11%2F2%2B%285sqrt%285%29%29%2F2\"\"%22%22=%22%22\"\"11\"\r\n" );
document.write( "\r\n" );
document.write( "(4) \"11%2F2-%285sqrt%285%29%29%2F2\"\"%22%22%2B%22%22\"\"11%2F2-%285sqrt%285%29%29%2F2\"\"%22%22=%22%22\"\"11-5sqrt%285%29\"\r\n" );
document.write( "\r\n" );
document.write( "Since (2) and (3) are the same, there are only three answers:\r\n" );
document.write( "\r\n" );
document.write( "\"11%2B5sqrt%285%29\", \"11\", and \"11-5sqrt%285%29\"\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );