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Show that the product of \(a\sqrt{b}+c\sqrt{d}\) and \(a\sqrt{b}-c\sqrt{d}\) is always rational if \(a,b,c\) and \(d\) are rational.

 May 10, 2018
 #1
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___ The product of   a√b + c√d   and   a√b - c√d
=              
  ( a√b + c√d )( a√b - c√d )
=   ___   ___   ___  
  ( a√b )( a√b ) + ( a√b )( -c√d ) + ( c√d )( a√b ) + ( c√d)(-c√d )
=              
  a2√b2 - ac√b√d + ac√b√d - c2√d2
=              
  a2√b2  -  c2√d2
=              
 

a2b  -  c2d

 

And    a2b  -  c2d   is always rational if  a ,  b ,  c , and  d  are rational.

 May 10, 2018

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