document.write( "Question 398063: Given that \"+sqrt%28+6.4%2A10%5En+%29+\" is a rational number, what can you say about the value of n?
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Algebra.Com's Answer #282205 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Given that \"+sqrt%28+6.4%2A10%5En+%29+\" is a rational number, what can you say about the value of n?
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document.write( "\"sqrt%286.4%2A10%5En%29\"\r\n" );
document.write( "\"%286.4%2A10%5En%29%5E%281%2F2%29\"\r\n" );
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document.write( "\"%28expr%2864%2F10%29%2A10%5En%29%5E%281%2F2%29\"\r\n" );
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document.write( "\"%2864%2A10%5E%28n-1%29%29%5E%281%2F2%29\"\r\n" );
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document.write( "\"%288%5E2%2A10%5E%28n-1%29%29%5E%281%2F2%29\"\r\n" );
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document.write( "\"drawing%28100%2C80%2C0%2C1%2C-1%2C1%2Clocate%280%2C0%2C8%2A10%5E%28%28n-1%29%2F2%29%29%29%29\"\r\n" );
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document.write( "Only integer powers of 10 are rational. So the exponent\r\n" );
document.write( "must be an integer, say k\r\n" );
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document.write( "\"%28n-1%29%2F2=k\"\r\n" );
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document.write( "n-1 = 2k\r\n" );
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document.write( "n = 2k+1\r\n" );
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document.write( "Therefore n must be odd in order that \"sqrt%286.4%2A10%5En%29\"\r\n" );
document.write( "be rational.\r\n" );
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\n" ); document.write( "What if both \"+sqrt%28+6.4%2A10%5En+%29+\" and \"+root%283%2C+6.4%2A10%5En+%29+\" are rational? What can you say about the value of n?
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document.write( "For\r\n" );
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document.write( "\"root%283%2C6.4%2A10%5En%29\"\r\n" );
document.write( "\"%286.4%2A10%5En%29%5E%281%2F3%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28expr%2864%2F10%29%2A10%5En%29%5E%281%2F3%29\"\r\n" );
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document.write( "\"%2864%2A10%5E%28n-1%29%29%5E%281%2F3%29\"\r\n" );
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document.write( "\"%284%5E3%2A10%5E%28n-1%29%29%5E%281%2F3%29\"\r\n" );
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document.write( "\"drawing%28100%2C80%2C0%2C1%2C-1%2C1%2Clocate%280%2C0%2C4%2A10%5E%28%28n-1%29%2F3%29%29%29%29\"\r\n" );
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document.write( "Only integer powers of 10 are rational. So the exponent\r\n" );
document.write( "must be an integer, say p\r\n" );
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document.write( "\"%28n-1%29%2F3=p\"\r\n" );
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document.write( "n-1 = 3p\r\n" );
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document.write( "n = 3p+1\r\n" );
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document.write( "Therefore n must be 1 more than a multiple of 3\r\n" );
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document.write( "Now we set the exponents equal:\r\n" );
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document.write( "n = 3p+1 = 2k+1\r\n" );
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document.write( "3p = 2k\r\n" );
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document.write( "2p+p = 2k\r\n" );
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document.write( "p + \"p%2F2\" = 2k\r\n" );
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document.write( "\"p%2F2\" = 2k-p\r\n" );
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document.write( "The right side is an integer, so the left side must be also\r\n" );
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document.write( "Let that integer be A\r\n" );
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document.write( "Then \r\n" );
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document.write( "\"p%2F2\" = A \r\n" );
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document.write( "p = 2A\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "3p = 2k\r\n" );
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document.write( "3(2A) = 2k\r\n" );
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document.write( "6A = 2k\r\n" );
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document.write( "3A = k\r\n" );
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document.write( "Now supstitute in\r\n" );
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document.write( "n = 3p+1 = 2k+1\r\n" );
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document.write( "n = 3(2A)+1 = 2(3A)+1\r\n" );
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document.write( "n = 6A+1 = 6A+1 \r\n" );
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document.write( "So n must be 1 more than a multiple of 6.\r\n" );
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document.write( "Edwin
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