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Question
In the following, find the values of a and b:
`(5 + 2sqrt(3))/(7 + 4sqrt(3)) = "a" + "b"sqrt(3)`
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Solution
`(5 + 2sqrt(3))/(7 + 4sqrt(3)`
= `(5 + 2sqrt(3))/(7 + 4sqrt(3)) xx (7 - 4sqrt(3))/(7 - 4sqrt(3)`
= `(5(7 - 4sqrt(3)) + 2sqrt(3)(7 - 4sqrt(3)))/((7)^2 - (4sqrt(3))^2`
= `(35 - 20sqrt(3) + 14sqrt(3) - 24)/(49 - 48)`
= `(11 - 6sqrt(3))/(1)`
= `11 + (-6)sqrt(3)`
= `"a" + "b"sqrt(3)`
Hence, a = 11 and b = -6
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