help proving.....

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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.

ant101 May 10, 2018

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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.

hectictar May 10, 2018

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