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Question
Simplify the following :
`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2)`
Solution
`(7sqrt(3))/(sqrt(10) + sqrt(3)) - (2sqrt(5))/(sqrt(6) + sqrt(5)) - (3sqrt(2))/(sqrt(15) + 3sqrt(2)`
Rationalizing the denominator of each term, we have
= `(7sqrt(3)(sqrt(10) - sqrt(3)))/((sqrt(10) + sqrt(3))(sqrt(10) - sqrt(3))) - (2sqrt(5)(sqrt(6) - sqrt(5)))/((sqrt(6) + sqrt(5))(sqrt(6) - sqrt(5))) - (3sqrt(2)(3sqrt(2) - sqrt(15)))/((sqrt(15) + 3sqrt(2))(3sqrt(2)-sqrt(15))`
= `(7sqrt(3)(sqrt(10)- sqrt(3)))/(10 - 3) - (2sqrt(5)(sqrt(6) - sqrt(5)))/(6 - 5) - (3sqrt(2)(3sqrt(2) - sqrt(15)))/(18 - 15)`
= `sqrt3(sqrt10 - sqrt3) - 2sqrt5(sqrt6 - sqrt5) - sqrt2(3sqrt2 - sqrt15)`
= `sqrt30 - 3 - 2sqrt30 + 10 - 6 + sqrt30`
= 0 − 9 + 10
= 1
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