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Question
Examine, whether the following number are rational or irrational:
`2+sqrt3`
Solution
Let `x=2+sqrt3` be the rational
squaring on both sides
`rArrx^2=(2+sqer3)^2`
`rArrx^2=4+3+4sqrt3`
`rArrx^2=7+4sqrt3`
`rArrx^2-7=4sqrt3`
`rArr(x^2-7)/4=sqrt3`
Since, x is rational
⇒ x2 is rational
⇒ x2 - 7 is rational
`rArr(x^2-7)/4` is rational
`rArr sqrt3` is rational
But, `sqrt3` is irrational
So, we arrive at a contradiction.
Hence, `2+sqrt3` is an irrational number.
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