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Simplify the expression
\[\frac{1}{\sqrt{36} + \sqrt{27}} + \frac{1}{\sqrt{27} + \sqrt{18}} + \frac{1}{\sqrt{18} + \sqrt{9}}.\]

 Apr 12, 2020
 #1
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please I really need help!

 Apr 12, 2020
 #2
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1 / [ sqrt(36) + sqrt(27) ]  =  1 / [ 6 + 3sqrt(3) ]            =  1 / [ 3·( 2 + sqrt(3) ) ]

1 / [ sqrt(27) + sqrt(18) ]  =  1 / [ 3sqrt(3) + 3sqrt(2) ]  =  1 / [ 3·( sqrt(3) + sqrt(2) ) ]

1 / [ sqrt(18) + sqrt(9) ]    =  1 / [ 3sqrt(2) + 3 ]            =  1 / [ 3·( sqrt(2) + 1 ) ]

 

So, the common denominator is:  3 · ( 2 + sqrt(3) ) · ( sqrt(3) + sqrt(2) ) ·( sqrt(2) + 1 ) ]

 

Multiply the numerator and denominator of the first term by:       ( sqrt(3) + sqrt(2) ) ·( sqrt(2) + 1 )

Multiply the numerator and denominator of the second term by:  ( 2 + sqrt(3) ) · ( sqrt(2) + 1 )

Multiply the numerator and denominator of the third term by:       ( 2 + sqrt(3) ) · ( sqrt(3) + sqrt(2) )

 

This will be messy; if you need further help -- ask ...

 Apr 12, 2020

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