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Express (3)/(sqrt(3) - sqrt(2) + sqrt(5)...

Express `(3)/(sqrt(3) - sqrt(2) + sqrt(5))` with rational denominator.

Text Solution

Verified by Experts

We have .
`(3)/((sqrt(3) - sqrt(2))+sqrt(5)) = (3)/((sqrt(3) - sqrt(2))+sqrt(5)) xx ((sqrt(3) - sqrt(2))- sqrt(5))/((sqrt(3) - sqrt(2) - sqrt(5)))`
` = (3(sqrt(3) - sqrt(2) - sqrt(5)))/((sqrt(3) - sqrt(2))^(2) -(sqrt(5))^(2)) = (3(sqrt(3) - sqrt(2) - sqrt(5)))/([(sqrt(3))^(2) + (sqrt(2))^(2) - 2 xx sqrt(3) xx sqrt(2)] - 5)`
`= (3(sqrt(3) - sqrt(2) - sqrt(5)))/((3+ 2 -2 sqrt(6)) -5) = (3(sqrt(3) - sqrt(2) - sqrt(5)))/(- 2sqrt(6)) xx (sqrt(6))/(sqrt(6))`
` = (3(sqrt(18) - sqrt(12) - sqrt(30)))/(-2xx6) = ( 3(sqrt(2) - 2sqrt(3) - sqrt(30)))/(-12)`
`= (3sqrt(2) - 2sqrt(3) - sqrt(30))/(-4) =(( 2sqrt(3) + sqrt(30) - 3sqrt(2)))/(4)`
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