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Question
In the following determine rational numbers a and b:
`(sqrt3 - 1)/(sqrt3 + 1) = a - bsqrt3`
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Solution
We know that rationalization factor for `sqrt3 + 1` is `sqrt3 - 1`. We will multiply numerator and denominator of the given expression `(sqrt3 - 1)/(sqrt3 + 1)` by `sqrt3 - 1` to get
`(sqrt3 - 1)/(sqrt3 + 1) xx (sqrt3 - 1)/(sqrt3 - 1) = ((sqrt3)^2 + (1)^2 - 2 xx sqrt3 xx 1)/((sqrt3)^2 - (1)^2)`
`= (3 + 1 - 2sqrt3)/(3 - 2)`
`= (4 - 2sqrt3`)/2`
`= 2 - sqrt3`
On equating rational and irrational terms, we get
`a - bsqrt3 = 2 - sqrt3`
`= 2 - 1sqrt3`
Hence we get a = 2, b = 1
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