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Question
Prove that `sqrt3+sqrt5` is an irrational number.
Solution
Given that `sqrt3+sqrt5` is an irrational number
Now we have to prove `sqrt3+sqrt5` is an irrational number
Let `x=sqrt3+sqrt5` is a rational
Squaring on both sides
`rArrx^2=(sqrt3+sqrt5)^2`
`rArrx^2=(sqrt3)^2+(sqrt5)^2+2sqrt3xxsqrt5`
`rArrx^2=3+5+2sqrt15`
`rArrx^2=8+2sqrt15`
`rArr(x^2-8)/2=sqrt15`
Now x is rational
⇒ x2 is rational
`rArr(x^2-8)/2` is rational
`rArr sqrt15` is rational
But, `sqrt15` is an irrational
Thus we arrive at contradiction that `sqrt3+sqrt5` is a rational which is wrong.
Hence `sqrt3+sqrt5` is an irrational.
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