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Question
In the following determine rational numbers a and b:
`(4 + sqrt2)/(2 + sqrt2) = n - sqrtb`
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Solution
We know that rationalization factor for `2 + sqrt2` is `2 - sqrt2`. We will multiply numerator and denominator of the given expression `(4 + sqrt2)/(2 + sqrt2)` by `2 - sqrt2` to get
`(4 + sqrt2)/(2 + sqrt2) xx (2 - sqrt2)/(2 - sqrt2) = (4 xx 2 - 4 xx sqrt2 + 2 xx sqrt2 - (sqrt2)^2)/((2)^2 - (sqrt2)^2)`
`= (8 - 4sqrt2 + 2sqrt2 - 2)/(4 - 2)`
`= (6 - 2sqrt2)/2`
`= 3 - sqrt2`
On equating rational and irrational terms, we get
`a - sqrtb = 3 - sqrt2`
Hence we get a = 3, b = 2
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