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Question
Rationales the denominator and simplify:
`(2sqrt6 - sqrt5)/(3sqrt5 - 2sqrt6)`
Solution
We know that rationalization factor for `3sqrt5 - 2sqrt6` is `3sqrt5 + 2sqrt6` . We will multiply numerator and denominator of the given expression `(2sqrt6 - sqrt5)/(3sqrt5 - 2sqrt6)` by `3sqrt5 + 2sqrt6` to get
`(2sqrt6 - sqrt5)/(3sqrt5 - 2sqrt6) xx (3sqrt5 + 2sqrt6)/(2sqrt + 2 sqrt6) = (2xx 3 xx sqrt6 + sqrt5 + (2sqrt6)^2 - 3 xx (sqrt5)^2 - 2 xx sqrt5 xx sqrt6)/((3sqrt5)^2 - (2sqrt6)^2)`
`= (6sqrt(6 xx 5) + 4 xx 6 - 3 xx (sqrt5)^2 - 2 xx sqrt5 xx sqrt6)/(9 xx 5 - 4 xx 6)`
` = (6sqrt30 + 24 - 15 - 2sqrt30)/(45 - 24)`
`= (9 + 4sqrt30)/21`
Hence the given expression is simplified to `(9 + 4sqrt30)/21`
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