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Prove that of the Numbers `Sqrt(3) + Sqrt(5)` is Irrational: - Mathematics

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Prove that of the numbers `sqrt(3) + sqrt(5)` is irrational: 

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Solution

Let`sqrt(3) + sqrt(5)` be rational.
∴`sqrt(3) + sqrt(5)` = a, where a is rational.
∴ `sqrt(3) = a - sqrt(5)`               ….(1)
On squaring both sides of equation (1), we get
`3 = (a - sqrt(5))^2 = a^2 + 5 - 2sqrt(5a)`
⇒ `sqrt(5) = (a^2+2) /(2a)`
This is impossible because right-hand side is rational, whereas the left-hand side is irrational.
This is a contradiction.
Hence,`sqrt(3) + sqrt(5)`  is irrational.

Concept: Concept of Irrational Numbers
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