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# hard algebra

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If $$a=(\sqrt{3}+\sqrt{2})^{-3}$$ and $$b=(\sqrt{3}-\sqrt{2})^{-3}$$, find the value of 1/(a + 1) + 1/(b + 1).

Jun 27, 2020

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To make the reading simpler, let  x  =  ( sqrt(3) + sqrt(2) )   --->   a  =  x-3

and                                         let  y  =  ( sqrt(3) - sqrt(2) )   --->   b  =  y-3

This means that  x · y  =  ( sqrt(3) + sqrt(2) ) · ( sqrt(3) - sqrt(2) )  =  3 - 2  =  1

and  x3y3  =  ( xy )3  =  ( 1 ) =  1

1 / (a + 1)  =  1 / ( x-3 + 1 )  =  x3 / ( 1 + x3 )                (multiplying both the numerator and denominator by x3)

1 / (b + 1)  =  1 / ( y-3 + 1 )  =  y3 / ( 1 + y3 )                (multiplying both the numerator and denominator by y3)

1 / (a + 1)  +  1 / (b + 1)  =  x3 / ( 1 + x3 )   +   y3 / ( 1 + y3 )

using the common denominator of  ( 1 + x3 ) · ( 1 + y3 ) :

=  { [ x3( 1 + y3 ) ] + [ y3( 1 + x3 ) ] }  /  [ ( 1 + x3 )( 1 + y3 ) ]

multiplying out both the numberator and denominator:

[ x3 + x3y3 + y3 + x3y3 ] / [ 1 + x3 + y3 + x3y3 ]

since x3y3  =  1:                   [ x3 + y3 + 2 ] / [ x3 + y3 + 2 ]  =  1.

Jun 27, 2020