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Question
Prove that `sqrt(3) + sqrt(5)` is irrational.
Solution
Let us suppose `sqrt(3) + sqrt(5)` is rational.
Let `sqrt(3) + sqrt(5)` = a, where a is rational.
Therefore, `sqrt(3) = a - sqrt(5)`
On squaring both sides, we get
`(sqrt(3))^2 = (a - sqrt(5))^2`
`\implies` 3 = `a^2 + 5 - 2a sqrt(5)` ......[∵ (a – b)2 = a2 + b2 – 2ab]
`\implies` `2a sqrt(5) = a^2 + 2`
Therefore, `sqrt(5) = (a^2 + 2)/(2a)`, which is a contradiction as the right-hand side is rational number while `sqrt(5)` is irrational.
Hence, `sqrt(3) + sqrt(5)` is irrational.
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