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Question
Examine, whether the following number are rational or irrational:
`sqrt3+sqrt5`
Solution
Let `x=sqrt3+sqrt5` be the rational number
Squaring on both sides, we get
`rArrx^2=(sqrt3+sqrt5)^2`
`rArrx^2=3+5+2sqrt15`
`rArrx^2=8+2sqrt15`
`rArrx^2-8=2sqrt15`
`rArr(x^2-8)/2=sqrt15`
Now, x is rational number
⇒ x2 is rational number
⇒ x2 - 8 is rational number
`rArr (x^2 -8)/2` is rational number
`rArrsqrt15` is rational number
But `sqrt15` is an irrational number
So, we arrive at a contradiction
Hence `sqrt3+sqrt5` is an irrational number.
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