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Question
Examine, whether the following number are rational or irrational:
`sqrt3+sqrt2`
Solution
Let `x=sqrt3+sqrt2` be the rational number
Squaring on both sides, we get
`rArrx^2=(sqrt3+sqrt2)^2`
`rArrx^2=3+2+2sqrt6`
`rArrx^2=5+2sqrt6`
`rArrx^2-5=2sqrt6`
`rArr(x^2-5)/2=sqrt6`
Since, x is rational number
⇒ x2 is rational
⇒ x2 - 5 is rational
`rArr(x^2-5)/2` is rational number
`rArr sqrt6` is rational number
But, `sqrt6` is an irrational number
So, we arrive at contradiction
Hence, `sqrt3+sqrt2` is an irrational number.
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