document.write( "Question 252194: 1.prove that √3 is irrational\r
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\n" ); document.write( "\n" ); document.write( "b. n√b < n√a where n√ is the positive nth term\r
\n" ); document.write( "\n" ); document.write( "c.1/bⁿ>1/aⁿ
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Algebra.Com's Answer #184066 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
1.prove that √3 is irrational
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document.write( "Assume for contradiction that \r\n" );
document.write( "equals a common fraction  reduced to lowest terms.\r\n" );
document.write( "That is, suppose there are integers p and q\r\n" );
document.write( "with no common factors other than 1 such that\r\n" );
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document.write( "Square both sides:\r\n" );
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document.write( "Multiply both sides by \r\n" );
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document.write( "q is either even or odd.  Suppose q is even.\r\n" );
document.write( "Then q^2 is even. Then 3q^2 is even.  Therefore\r\n" );
document.write( "p^2 is even, and therefore p is even.  That\r\n" );
document.write( "contradicts the fact that  was reduced\r\n" );
document.write( "to lowest terms, since if both were even they\r\n" );
document.write( "would have factor 2 in common.\r\n" );
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document.write( "So we have ruled out q being even,  So let's\r\n" );
document.write( "suppose q is odd. Then q^2 is odd.  Therefore\r\n" );
document.write( "3q^2 is odd. Therefore p^2 is odd. Therefore\r\n" );
document.write( "p is odd.  So there must exist non-negative\r\n" );
document.write( "integers m and n such that \r\n" );
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document.write( "p = 2n+1 and q = 2m+1.  Substituting in\r\n" );
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document.write( "Squaring these out:\r\n" );
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document.write( "Divide through by 2:\r\n" );
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document.write( "The left side is odd but the \r\n" );
document.write( "right side is even.  That cannot\r\n" );
document.write( "be, so q is not odd.\r\n" );
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document.write( "q cannot be even or odd, which cannot\r\n" );
document.write( "be, so  is irrational.\r\n" );
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\n" ); document.write( "2.prove that if 0 < b < a and n is a positive integer,then \r
\n" ); document.write( "\n" ); document.write( "a. $\"b%5En%3Ca%5En\"$
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document.write( " is given,\r\n" );
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document.write( "therefore \r\n" );
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document.write( "by a factoring theorem\r\n" );
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document.write( "Since the second parentheses contains only positive terms,\r\n" );
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document.write( " which is the same as \r\n" );
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\n" ); document.write( "b. $\"root%28n%2Ca%29+%3C+root%28n%2Ca%29\"$ where $\"root%28n%2C%22%22%29\"$ is the positive nth root
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document.write( "This follows by replacing  and  respectively with\r\n" );
document.write( " and  in part a. \r\n" );
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\n" ); document.write( "c. $\"1%2Fb%5En%3E1%2Fa%5En\"$
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document.write( " by part a.  This is equivalent to:\r\n" );
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document.write( "Divide through by the positive number \r\n" );
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document.write( "which is equivalent to  \r\n" );
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document.write( "Edwin

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